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Theorem axdc3lem2 9595
Description: Lemma for axdc3 9598. We have constructed a "candidate set" 𝑆, which consists of all finite sequences 𝑠 that satisfy our property of interest, namely 𝑠(𝑥 + 1) ∈ 𝐹(𝑠(𝑥)) on its domain, but with the added constraint that 𝑠(0) = 𝐶. These sets are possible "initial segments" of the infinite sequence satisfying these constraints, but we can leverage the standard ax-dc 9590 (with no initial condition) to select a sequence of ever-lengthening finite sequences, namely (𝑛):𝑚𝐴 (for some integer 𝑚). We let our "choice" function select a sequence whose domain is one more than the last one, and agrees with the previous one on its domain. Thus, the application of vanilla ax-dc 9590 yields a sequence of sequences whose domains increase without bound, and whose union is a function which has all the properties we want. In this lemma, we show that given the sequence , we can construct the sequence 𝑔 that we are after. (Contributed by Mario Carneiro, 30-Jan-2013.)
Hypotheses
Ref Expression
axdc3lem2.1 𝐴 ∈ V
axdc3lem2.2 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
axdc3lem2.3 𝐺 = (𝑥𝑆 ↦ {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})
Assertion
Ref Expression
axdc3lem2 (∃(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
Distinct variable groups:   𝐴,𝑔,   𝐴,𝑛,𝑠   𝐶,𝑔,   𝐶,𝑛,𝑠   𝑔,𝐹,   𝑛,𝐹,𝑠   𝑘,𝐺   𝑆,𝑘,𝑠   𝑥,𝑆,𝑦   𝑔,𝑘,   ,𝑠   𝑥,,𝑦   𝑘,𝑛
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑘)   𝐶(𝑥,𝑦,𝑘)   𝑆(𝑔,,𝑛)   𝐹(𝑥,𝑦,𝑘)   𝐺(𝑥,𝑦,𝑔,,𝑛,𝑠)

Proof of Theorem axdc3lem2
Dummy variables 𝑖 𝑗 𝑚 𝑢 𝑣 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . . . . . . . 13 (𝑚 = ∅ → 𝑚 = ∅)
2 fveq2 6437 . . . . . . . . . . . . . 14 (𝑚 = ∅ → (𝑚) = (‘∅))
32dmeqd 5562 . . . . . . . . . . . . 13 (𝑚 = ∅ → dom (𝑚) = dom (‘∅))
41, 3eleq12d 2900 . . . . . . . . . . . 12 (𝑚 = ∅ → (𝑚 ∈ dom (𝑚) ↔ ∅ ∈ dom (‘∅)))
5 eleq2 2895 . . . . . . . . . . . . 13 (𝑚 = ∅ → (𝑗𝑚𝑗 ∈ ∅))
62sseq2d 3858 . . . . . . . . . . . . 13 (𝑚 = ∅ → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (‘∅)))
75, 6imbi12d 336 . . . . . . . . . . . 12 (𝑚 = ∅ → ((𝑗𝑚 → (𝑗) ⊆ (𝑚)) ↔ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))))
84, 7anbi12d 624 . . . . . . . . . . 11 (𝑚 = ∅ → ((𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))) ↔ (∅ ∈ dom (‘∅) ∧ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅)))))
9 id 22 . . . . . . . . . . . . 13 (𝑚 = 𝑖𝑚 = 𝑖)
10 fveq2 6437 . . . . . . . . . . . . . 14 (𝑚 = 𝑖 → (𝑚) = (𝑖))
1110dmeqd 5562 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → dom (𝑚) = dom (𝑖))
129, 11eleq12d 2900 . . . . . . . . . . . 12 (𝑚 = 𝑖 → (𝑚 ∈ dom (𝑚) ↔ 𝑖 ∈ dom (𝑖)))
13 elequ2 2178 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → (𝑗𝑚𝑗𝑖))
1410sseq2d 3858 . . . . . . . . . . . . 13 (𝑚 = 𝑖 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (𝑖)))
1513, 14imbi12d 336 . . . . . . . . . . . 12 (𝑚 = 𝑖 → ((𝑗𝑚 → (𝑗) ⊆ (𝑚)) ↔ (𝑗𝑖 → (𝑗) ⊆ (𝑖))))
1612, 15anbi12d 624 . . . . . . . . . . 11 (𝑚 = 𝑖 → ((𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))) ↔ (𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖)))))
17 id 22 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖𝑚 = suc 𝑖)
18 fveq2 6437 . . . . . . . . . . . . . 14 (𝑚 = suc 𝑖 → (𝑚) = (‘suc 𝑖))
1918dmeqd 5562 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → dom (𝑚) = dom (‘suc 𝑖))
2017, 19eleq12d 2900 . . . . . . . . . . . 12 (𝑚 = suc 𝑖 → (𝑚 ∈ dom (𝑚) ↔ suc 𝑖 ∈ dom (‘suc 𝑖)))
21 eleq2 2895 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → (𝑗𝑚𝑗 ∈ suc 𝑖))
2218sseq2d 3858 . . . . . . . . . . . . 13 (𝑚 = suc 𝑖 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (‘suc 𝑖)))
2321, 22imbi12d 336 . . . . . . . . . . . 12 (𝑚 = suc 𝑖 → ((𝑗𝑚 → (𝑗) ⊆ (𝑚)) ↔ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))
2420, 23anbi12d 624 . . . . . . . . . . 11 (𝑚 = suc 𝑖 → ((𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))) ↔ (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖)))))
25 peano1 7351 . . . . . . . . . . . . . . 15 ∅ ∈ ω
26 ffvelrn 6611 . . . . . . . . . . . . . . 15 ((:ω⟶𝑆 ∧ ∅ ∈ ω) → (‘∅) ∈ 𝑆)
2725, 26mpan2 682 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → (‘∅) ∈ 𝑆)
28 axdc3lem2.2 . . . . . . . . . . . . . . . . . 18 𝑆 = {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
29 fdm 6290 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠:suc 𝑛𝐴 → dom 𝑠 = suc 𝑛)
30 nnord 7339 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ω → Ord 𝑛)
31 0elsuc 7301 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Ord 𝑛 → ∅ ∈ suc 𝑛)
3230, 31syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ω → ∅ ∈ suc 𝑛)
33 peano2 7352 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ω → suc 𝑛 ∈ ω)
34 eleq2 2895 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑠 = suc 𝑛 → (∅ ∈ dom 𝑠 ↔ ∅ ∈ suc 𝑛))
35 eleq1 2894 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑠 = suc 𝑛 → (dom 𝑠 ∈ ω ↔ suc 𝑛 ∈ ω))
3634, 35anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . . . 25 (dom 𝑠 = suc 𝑛 → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ suc 𝑛 ∧ suc 𝑛 ∈ ω)))
3736biimprcd 242 . . . . . . . . . . . . . . . . . . . . . . . 24 ((∅ ∈ suc 𝑛 ∧ suc 𝑛 ∈ ω) → (dom 𝑠 = suc 𝑛 → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
3832, 33, 37syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ω → (dom 𝑠 = suc 𝑛 → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
3929, 38syl5com 31 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠:suc 𝑛𝐴 → (𝑛 ∈ ω → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
40393ad2ant1 1167 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑛 ∈ ω → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)))
4140impcom 398 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))) → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω))
4241rexlimiva 3237 . . . . . . . . . . . . . . . . . . 19 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω))
4342ss2abi 3901 . . . . . . . . . . . . . . . . . 18 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}
4428, 43eqsstri 3860 . . . . . . . . . . . . . . . . 17 𝑆 ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}
4544sseli 3823 . . . . . . . . . . . . . . . 16 ((‘∅) ∈ 𝑆 → (‘∅) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)})
46 fvex 6450 . . . . . . . . . . . . . . . . 17 (‘∅) ∈ V
47 dmeq 5560 . . . . . . . . . . . . . . . . . . 19 (𝑠 = (‘∅) → dom 𝑠 = dom (‘∅))
4847eleq2d 2892 . . . . . . . . . . . . . . . . . 18 (𝑠 = (‘∅) → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom (‘∅)))
4947eleq1d 2891 . . . . . . . . . . . . . . . . . 18 (𝑠 = (‘∅) → (dom 𝑠 ∈ ω ↔ dom (‘∅) ∈ ω))
5048, 49anbi12d 624 . . . . . . . . . . . . . . . . 17 (𝑠 = (‘∅) → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom (‘∅) ∧ dom (‘∅) ∈ ω)))
5146, 50elab 3571 . . . . . . . . . . . . . . . 16 ((‘∅) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom (‘∅) ∧ dom (‘∅) ∈ ω))
5245, 51sylib 210 . . . . . . . . . . . . . . 15 ((‘∅) ∈ 𝑆 → (∅ ∈ dom (‘∅) ∧ dom (‘∅) ∈ ω))
5352simpld 490 . . . . . . . . . . . . . 14 ((‘∅) ∈ 𝑆 → ∅ ∈ dom (‘∅))
5427, 53syl 17 . . . . . . . . . . . . 13 (:ω⟶𝑆 → ∅ ∈ dom (‘∅))
55 noel 4150 . . . . . . . . . . . . . 14 ¬ 𝑗 ∈ ∅
5655pm2.21i 117 . . . . . . . . . . . . 13 (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))
5754, 56jctir 516 . . . . . . . . . . . 12 (:ω⟶𝑆 → (∅ ∈ dom (‘∅) ∧ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))))
5857adantr 474 . . . . . . . . . . 11 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (∅ ∈ dom (‘∅) ∧ (𝑗 ∈ ∅ → (𝑗) ⊆ (‘∅))))
59 ffvelrn 6611 . . . . . . . . . . . . . . 15 ((:ω⟶𝑆𝑖 ∈ ω) → (𝑖) ∈ 𝑆)
6059ancoms 452 . . . . . . . . . . . . . 14 ((𝑖 ∈ ω ∧ :ω⟶𝑆) → (𝑖) ∈ 𝑆)
6160adantrr 708 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑖) ∈ 𝑆)
62 suceq 6032 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑖 → suc 𝑘 = suc 𝑖)
6362fveq2d 6441 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑖 → (‘suc 𝑘) = (‘suc 𝑖))
64 2fveq3 6442 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑖 → (𝐺‘(𝑘)) = (𝐺‘(𝑖)))
6563, 64eleq12d 2900 . . . . . . . . . . . . . . 15 (𝑘 = 𝑖 → ((‘suc 𝑘) ∈ (𝐺‘(𝑘)) ↔ (‘suc 𝑖) ∈ (𝐺‘(𝑖))))
6665rspcva 3524 . . . . . . . . . . . . . 14 ((𝑖 ∈ ω ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (‘suc 𝑖) ∈ (𝐺‘(𝑖)))
6766adantrl 707 . . . . . . . . . . . . 13 ((𝑖 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (‘suc 𝑖) ∈ (𝐺‘(𝑖)))
6844sseli 3823 . . . . . . . . . . . . . . . . . . . 20 ((𝑖) ∈ 𝑆 → (𝑖) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)})
69 fvex 6450 . . . . . . . . . . . . . . . . . . . . 21 (𝑖) ∈ V
70 dmeq 5560 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 = (𝑖) → dom 𝑠 = dom (𝑖))
7170eleq2d 2892 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = (𝑖) → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom (𝑖)))
7270eleq1d 2891 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 = (𝑖) → (dom 𝑠 ∈ ω ↔ dom (𝑖) ∈ ω))
7371, 72anbi12d 624 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 = (𝑖) → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom (𝑖) ∧ dom (𝑖) ∈ ω)))
7469, 73elab 3571 . . . . . . . . . . . . . . . . . . . 20 ((𝑖) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom (𝑖) ∧ dom (𝑖) ∈ ω))
7568, 74sylib 210 . . . . . . . . . . . . . . . . . . 19 ((𝑖) ∈ 𝑆 → (∅ ∈ dom (𝑖) ∧ dom (𝑖) ∈ ω))
7675simprd 491 . . . . . . . . . . . . . . . . . 18 ((𝑖) ∈ 𝑆 → dom (𝑖) ∈ ω)
77 nnord 7339 . . . . . . . . . . . . . . . . . 18 (dom (𝑖) ∈ ω → Ord dom (𝑖))
78 ordsucelsuc 7288 . . . . . . . . . . . . . . . . . 18 (Ord dom (𝑖) → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
7976, 77, 783syl 18 . . . . . . . . . . . . . . . . 17 ((𝑖) ∈ 𝑆 → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
8079adantr 474 . . . . . . . . . . . . . . . 16 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
81 dmeq 5560 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = (𝑖) → dom 𝑥 = dom (𝑖))
82 suceq 6032 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑥 = dom (𝑖) → suc dom 𝑥 = suc dom (𝑖))
8381, 82syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑖) → suc dom 𝑥 = suc dom (𝑖))
8483eqeq2d 2835 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑖) → (dom 𝑦 = suc dom 𝑥 ↔ dom 𝑦 = suc dom (𝑖)))
8581reseq2d 5633 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑖) → (𝑦 ↾ dom 𝑥) = (𝑦 ↾ dom (𝑖)))
86 id 22 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = (𝑖) → 𝑥 = (𝑖))
8785, 86eqeq12d 2840 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = (𝑖) → ((𝑦 ↾ dom 𝑥) = 𝑥 ↔ (𝑦 ↾ dom (𝑖)) = (𝑖)))
8884, 87anbi12d 624 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = (𝑖) → ((dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥) ↔ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))))
8988rabbidv 3402 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = (𝑖) → {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)} = {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))})
90 axdc3lem2.3 . . . . . . . . . . . . . . . . . . . . . 22 𝐺 = (𝑥𝑆 ↦ {𝑦𝑆 ∣ (dom 𝑦 = suc dom 𝑥 ∧ (𝑦 ↾ dom 𝑥) = 𝑥)})
91 axdc3lem2.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝐴 ∈ V
9291, 28axdc3lem 9594 . . . . . . . . . . . . . . . . . . . . . . 23 𝑆 ∈ V
9392rabex 5039 . . . . . . . . . . . . . . . . . . . . . 22 {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))} ∈ V
9489, 90, 93fvmpt 6533 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖) ∈ 𝑆 → (𝐺‘(𝑖)) = {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))})
9594eleq2d 2892 . . . . . . . . . . . . . . . . . . . 20 ((𝑖) ∈ 𝑆 → ((‘suc 𝑖) ∈ (𝐺‘(𝑖)) ↔ (‘suc 𝑖) ∈ {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))}))
96 dmeq 5560 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (‘suc 𝑖) → dom 𝑦 = dom (‘suc 𝑖))
9796eqeq1d 2827 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (‘suc 𝑖) → (dom 𝑦 = suc dom (𝑖) ↔ dom (‘suc 𝑖) = suc dom (𝑖)))
98 reseq1 5627 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = (‘suc 𝑖) → (𝑦 ↾ dom (𝑖)) = ((‘suc 𝑖) ↾ dom (𝑖)))
9998eqeq1d 2827 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = (‘suc 𝑖) → ((𝑦 ↾ dom (𝑖)) = (𝑖) ↔ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖)))
10097, 99anbi12d 624 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = (‘suc 𝑖) → ((dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖)) ↔ (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖))))
101100elrab 3585 . . . . . . . . . . . . . . . . . . . 20 ((‘suc 𝑖) ∈ {𝑦𝑆 ∣ (dom 𝑦 = suc dom (𝑖) ∧ (𝑦 ↾ dom (𝑖)) = (𝑖))} ↔ ((‘suc 𝑖) ∈ 𝑆 ∧ (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖))))
10295, 101syl6bb 279 . . . . . . . . . . . . . . . . . . 19 ((𝑖) ∈ 𝑆 → ((‘suc 𝑖) ∈ (𝐺‘(𝑖)) ↔ ((‘suc 𝑖) ∈ 𝑆 ∧ (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖)))))
103102simplbda 495 . . . . . . . . . . . . . . . . . 18 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (dom (‘suc 𝑖) = suc dom (𝑖) ∧ ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖)))
104103simpld 490 . . . . . . . . . . . . . . . . 17 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → dom (‘suc 𝑖) = suc dom (𝑖))
105104eleq2d 2892 . . . . . . . . . . . . . . . 16 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ↔ suc 𝑖 ∈ suc dom (𝑖)))
10680, 105bitr4d 274 . . . . . . . . . . . . . . 15 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (𝑖 ∈ dom (𝑖) ↔ suc 𝑖 ∈ dom (‘suc 𝑖)))
107106biimpd 221 . . . . . . . . . . . . . 14 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → (𝑖 ∈ dom (𝑖) → suc 𝑖 ∈ dom (‘suc 𝑖)))
108103simprd 491 . . . . . . . . . . . . . . 15 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → ((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖))
109 resss 5662 . . . . . . . . . . . . . . . 16 ((‘suc 𝑖) ↾ dom (𝑖)) ⊆ (‘suc 𝑖)
110 sseq1 3851 . . . . . . . . . . . . . . . 16 (((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖) → (((‘suc 𝑖) ↾ dom (𝑖)) ⊆ (‘suc 𝑖) ↔ (𝑖) ⊆ (‘suc 𝑖)))
111109, 110mpbii 225 . . . . . . . . . . . . . . 15 (((‘suc 𝑖) ↾ dom (𝑖)) = (𝑖) → (𝑖) ⊆ (‘suc 𝑖))
112 elsuci 6033 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ suc 𝑖 → (𝑗𝑖𝑗 = 𝑖))
113 pm2.27 42 . . . . . . . . . . . . . . . . . . 19 (𝑗𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → (𝑗) ⊆ (𝑖)))
114 sstr2 3834 . . . . . . . . . . . . . . . . . . 19 ((𝑗) ⊆ (𝑖) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖)))
115113, 114syl6 35 . . . . . . . . . . . . . . . . . 18 (𝑗𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
116 fveq2 6437 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 = 𝑖 → (𝑗) = (𝑖))
117116sseq1d 3857 . . . . . . . . . . . . . . . . . . . 20 (𝑗 = 𝑖 → ((𝑗) ⊆ (‘suc 𝑖) ↔ (𝑖) ⊆ (‘suc 𝑖)))
118117biimprd 240 . . . . . . . . . . . . . . . . . . 19 (𝑗 = 𝑖 → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖)))
119118a1d 25 . . . . . . . . . . . . . . . . . 18 (𝑗 = 𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
120115, 119jaoi 888 . . . . . . . . . . . . . . . . 17 ((𝑗𝑖𝑗 = 𝑖) → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
121112, 120syl 17 . . . . . . . . . . . . . . . 16 (𝑗 ∈ suc 𝑖 → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → ((𝑖) ⊆ (‘suc 𝑖) → (𝑗) ⊆ (‘suc 𝑖))))
122121com13 88 . . . . . . . . . . . . . . 15 ((𝑖) ⊆ (‘suc 𝑖) → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))
123108, 111, 1223syl 18 . . . . . . . . . . . . . 14 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → ((𝑗𝑖 → (𝑗) ⊆ (𝑖)) → (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))
124107, 123anim12d 602 . . . . . . . . . . . . 13 (((𝑖) ∈ 𝑆 ∧ (‘suc 𝑖) ∈ (𝐺‘(𝑖))) → ((𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖)))))
12561, 67, 124syl2anc 579 . . . . . . . . . . . 12 ((𝑖 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → ((𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖)))))
126125ex 403 . . . . . . . . . . 11 (𝑖 ∈ ω → ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ((𝑖 ∈ dom (𝑖) ∧ (𝑗𝑖 → (𝑗) ⊆ (𝑖))) → (suc 𝑖 ∈ dom (‘suc 𝑖) ∧ (𝑗 ∈ suc 𝑖 → (𝑗) ⊆ (‘suc 𝑖))))))
1278, 16, 24, 58, 126finds2 7360 . . . . . . . . . 10 (𝑚 ∈ ω → ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚)))))
128127imp 397 . . . . . . . . 9 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑚 ∈ dom (𝑚) ∧ (𝑗𝑚 → (𝑗) ⊆ (𝑚))))
129128simprd 491 . . . . . . . 8 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑗𝑚 → (𝑗) ⊆ (𝑚)))
130129expcom 404 . . . . . . 7 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → (𝑗𝑚 → (𝑗) ⊆ (𝑚))))
131130ralrimdv 3177 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → ∀𝑗𝑚 (𝑗) ⊆ (𝑚)))
132131ralrimiv 3174 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚))
133 frn 6288 . . . . . . . . . . . 12 (:ω⟶𝑆 → ran 𝑆)
134 ffun 6285 . . . . . . . . . . . . . . . 16 (𝑠:suc 𝑛𝐴 → Fun 𝑠)
1351343ad2ant1 1167 . . . . . . . . . . . . . . 15 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → Fun 𝑠)
136135rexlimivw 3238 . . . . . . . . . . . . . 14 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → Fun 𝑠)
137136ss2abi 3901 . . . . . . . . . . . . 13 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ Fun 𝑠}
13828, 137eqsstri 3860 . . . . . . . . . . . 12 𝑆 ⊆ {𝑠 ∣ Fun 𝑠}
139133, 138syl6ss 3839 . . . . . . . . . . 11 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ Fun 𝑠})
140139sseld 3826 . . . . . . . . . 10 (:ω⟶𝑆 → (𝑢 ∈ ran 𝑢 ∈ {𝑠 ∣ Fun 𝑠}))
141 vex 3417 . . . . . . . . . . 11 𝑢 ∈ V
142 funeq 6147 . . . . . . . . . . 11 (𝑠 = 𝑢 → (Fun 𝑠 ↔ Fun 𝑢))
143141, 142elab 3571 . . . . . . . . . 10 (𝑢 ∈ {𝑠 ∣ Fun 𝑠} ↔ Fun 𝑢)
144140, 143syl6ib 243 . . . . . . . . 9 (:ω⟶𝑆 → (𝑢 ∈ ran → Fun 𝑢))
145144adantr 474 . . . . . . . 8 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → Fun 𝑢))
146 ffn 6282 . . . . . . . . 9 (:ω⟶𝑆 Fn ω)
147 fvelrnb 6494 . . . . . . . . . . . . 13 ( Fn ω → (𝑣 ∈ ran ↔ ∃𝑏 ∈ ω (𝑏) = 𝑣))
148 fvelrnb 6494 . . . . . . . . . . . . . . 15 ( Fn ω → (𝑢 ∈ ran ↔ ∃𝑎 ∈ ω (𝑎) = 𝑢))
149 nnord 7339 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑎 ∈ ω → Ord 𝑎)
150 nnord 7339 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 ∈ ω → Ord 𝑏)
151149, 150anim12i 606 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (Ord 𝑎 ∧ Ord 𝑏))
152 ordtri3or 5999 . . . . . . . . . . . . . . . . . . . . . . 23 ((Ord 𝑎 ∧ Ord 𝑏) → (𝑎𝑏𝑎 = 𝑏𝑏𝑎))
153 fveq2 6437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 = 𝑏 → (𝑚) = (𝑏))
154153sseq2d 3858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 = 𝑏 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (𝑏)))
155154raleqbi1dv 3358 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 = 𝑏 → (∀𝑗𝑚 (𝑗) ⊆ (𝑚) ↔ ∀𝑗𝑏 (𝑗) ⊆ (𝑏)))
156155rspcv 3522 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ∀𝑗𝑏 (𝑗) ⊆ (𝑏)))
157 fveq2 6437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑎 → (𝑗) = (𝑎))
158157sseq1d 3857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑎 → ((𝑗) ⊆ (𝑏) ↔ (𝑎) ⊆ (𝑏)))
159158rspccv 3523 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑗𝑏 (𝑗) ⊆ (𝑏) → (𝑎𝑏 → (𝑎) ⊆ (𝑏)))
160156, 159syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑎𝑏 → (𝑎) ⊆ (𝑏))))
161160adantl 475 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑎𝑏 → (𝑎) ⊆ (𝑏))))
1621613imp 1141 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑎𝑏) → (𝑎) ⊆ (𝑏))
163162orcd 904 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑎𝑏) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
1641633exp 1152 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑎𝑏 → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
165164com3r 87 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎𝑏 → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
166 fveq2 6437 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎 = 𝑏 → (𝑎) = (𝑏))
167 eqimss 3882 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑎) = (𝑏) → (𝑎) ⊆ (𝑏))
168167orcd 904 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎) = (𝑏) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
169166, 168syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎 = 𝑏 → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
1701692a1d 26 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑎 = 𝑏 → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
171 fveq2 6437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 = 𝑎 → (𝑚) = (𝑎))
172171sseq2d 3858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚 = 𝑎 → ((𝑗) ⊆ (𝑚) ↔ (𝑗) ⊆ (𝑎)))
173172raleqbi1dv 3358 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑚 = 𝑎 → (∀𝑗𝑚 (𝑗) ⊆ (𝑚) ↔ ∀𝑗𝑎 (𝑗) ⊆ (𝑎)))
174173rspcv 3522 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑎 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ∀𝑗𝑎 (𝑗) ⊆ (𝑎)))
175 fveq2 6437 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑏 → (𝑗) = (𝑏))
176175sseq1d 3857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑏 → ((𝑗) ⊆ (𝑎) ↔ (𝑏) ⊆ (𝑎)))
177176rspccv 3523 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∀𝑗𝑎 (𝑗) ⊆ (𝑎) → (𝑏𝑎 → (𝑏) ⊆ (𝑎)))
178174, 177syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑎 ∈ ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑏𝑎 → (𝑏) ⊆ (𝑎))))
179178adantr 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑏𝑎 → (𝑏) ⊆ (𝑎))))
1801793imp 1141 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑏𝑎) → (𝑏) ⊆ (𝑎))
181180olcd 905 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) ∧ 𝑏𝑎) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))
1821813exp 1152 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑏𝑎 → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
183182com3r 87 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏𝑎 → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
184165, 170, 1833jaoi 1556 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎𝑏𝑎 = 𝑏𝑏𝑎) → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
185152, 184syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((Ord 𝑎 ∧ Ord 𝑏) → ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)))))
186151, 185mpcom 38 . . . . . . . . . . . . . . . . . . . . 21 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → ((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎))))
187 sseq12 3853 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → ((𝑎) ⊆ (𝑏) ↔ 𝑢𝑣))
188 sseq12 3853 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑏) = 𝑣 ∧ (𝑎) = 𝑢) → ((𝑏) ⊆ (𝑎) ↔ 𝑣𝑢))
189188ancoms 452 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → ((𝑏) ⊆ (𝑎) ↔ 𝑣𝑢))
190187, 189orbi12d 947 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)) ↔ (𝑢𝑣𝑣𝑢)))
191190biimpcd 241 . . . . . . . . . . . . . . . . . . . . 21 (((𝑎) ⊆ (𝑏) ∨ (𝑏) ⊆ (𝑎)) → (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (𝑢𝑣𝑣𝑢)))
192186, 191syl6 35 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (𝑢𝑣𝑣𝑢))))
193192com23 86 . . . . . . . . . . . . . . . . . . 19 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (((𝑎) = 𝑢 ∧ (𝑏) = 𝑣) → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))
194193exp4b 423 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ ω → (𝑏 ∈ ω → ((𝑎) = 𝑢 → ((𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))))
195194com23 86 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ ω → ((𝑎) = 𝑢 → (𝑏 ∈ ω → ((𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))))
196195rexlimiv 3236 . . . . . . . . . . . . . . . 16 (∃𝑎 ∈ ω (𝑎) = 𝑢 → (𝑏 ∈ ω → ((𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
197196rexlimdv 3239 . . . . . . . . . . . . . . 15 (∃𝑎 ∈ ω (𝑎) = 𝑢 → (∃𝑏 ∈ ω (𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢))))
198148, 197syl6bi 245 . . . . . . . . . . . . . 14 ( Fn ω → (𝑢 ∈ ran → (∃𝑏 ∈ ω (𝑏) = 𝑣 → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
199198com23 86 . . . . . . . . . . . . 13 ( Fn ω → (∃𝑏 ∈ ω (𝑏) = 𝑣 → (𝑢 ∈ ran → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
200147, 199sylbid 232 . . . . . . . . . . . 12 ( Fn ω → (𝑣 ∈ ran → (𝑢 ∈ ran → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢𝑣𝑣𝑢)))))
201200com24 95 . . . . . . . . . . 11 ( Fn ω → (∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚) → (𝑢 ∈ ran → (𝑣 ∈ ran → (𝑢𝑣𝑣𝑢)))))
202201imp 397 . . . . . . . . . 10 (( Fn ω ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → (𝑣 ∈ ran → (𝑢𝑣𝑣𝑢))))
203202ralrimdv 3177 . . . . . . . . 9 (( Fn ω ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)))
204146, 203sylan 575 . . . . . . . 8 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)))
205145, 204jcad 508 . . . . . . 7 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → (𝑢 ∈ ran → (Fun 𝑢 ∧ ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢))))
206205ralrimiv 3174 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → ∀𝑢 ∈ ran (Fun 𝑢 ∧ ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)))
207 fununi 6201 . . . . . 6 (∀𝑢 ∈ ran (Fun 𝑢 ∧ ∀𝑣 ∈ ran (𝑢𝑣𝑣𝑢)) → Fun ran )
208206, 207syl 17 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑚 ∈ ω ∀𝑗𝑚 (𝑗) ⊆ (𝑚)) → Fun ran )
209132, 208syldan 585 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → Fun ran )
210 vex 3417 . . . . . . . . 9 𝑚 ∈ V
211210eldm2 5558 . . . . . . . 8 (𝑚 ∈ dom ran ↔ ∃𝑢𝑚, 𝑢⟩ ∈ ran )
212 eluni2 4664 . . . . . . . . . 10 (⟨𝑚, 𝑢⟩ ∈ ran ↔ ∃𝑣 ∈ ran 𝑚, 𝑢⟩ ∈ 𝑣)
213210, 141opeldm 5564 . . . . . . . . . . . . . . 15 (⟨𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ dom 𝑣)
214213a1i 11 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → (⟨𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ dom 𝑣))
215133, 44syl6ss 3839 . . . . . . . . . . . . . . 15 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)})
216 ssel 3821 . . . . . . . . . . . . . . . 16 (ran ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} → (𝑣 ∈ ran 𝑣 ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}))
217 vex 3417 . . . . . . . . . . . . . . . . . 18 𝑣 ∈ V
218 dmeq 5560 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑣 → dom 𝑠 = dom 𝑣)
219218eleq2d 2892 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑣 → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom 𝑣))
220218eleq1d 2891 . . . . . . . . . . . . . . . . . . 19 (𝑠 = 𝑣 → (dom 𝑠 ∈ ω ↔ dom 𝑣 ∈ ω))
221219, 220anbi12d 624 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑣 → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω)))
222217, 221elab 3571 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω))
223222simprbi 492 . . . . . . . . . . . . . . . 16 (𝑣 ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} → dom 𝑣 ∈ ω)
224216, 223syl6 35 . . . . . . . . . . . . . . 15 (ran ⊆ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} → (𝑣 ∈ ran → dom 𝑣 ∈ ω))
225215, 224syl 17 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → (𝑣 ∈ ran → dom 𝑣 ∈ ω))
226214, 225anim12d 602 . . . . . . . . . . . . 13 (:ω⟶𝑆 → ((⟨𝑚, 𝑢⟩ ∈ 𝑣𝑣 ∈ ran ) → (𝑚 ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω)))
227 elnn 7341 . . . . . . . . . . . . 13 ((𝑚 ∈ dom 𝑣 ∧ dom 𝑣 ∈ ω) → 𝑚 ∈ ω)
228226, 227syl6 35 . . . . . . . . . . . 12 (:ω⟶𝑆 → ((⟨𝑚, 𝑢⟩ ∈ 𝑣𝑣 ∈ ran ) → 𝑚 ∈ ω))
229228expcomd 408 . . . . . . . . . . 11 (:ω⟶𝑆 → (𝑣 ∈ ran → (⟨𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ ω)))
230229rexlimdv 3239 . . . . . . . . . 10 (:ω⟶𝑆 → (∃𝑣 ∈ ran 𝑚, 𝑢⟩ ∈ 𝑣𝑚 ∈ ω))
231212, 230syl5bi 234 . . . . . . . . 9 (:ω⟶𝑆 → (⟨𝑚, 𝑢⟩ ∈ ran 𝑚 ∈ ω))
232231exlimdv 2032 . . . . . . . 8 (:ω⟶𝑆 → (∃𝑢𝑚, 𝑢⟩ ∈ ran 𝑚 ∈ ω))
233211, 232syl5bi 234 . . . . . . 7 (:ω⟶𝑆 → (𝑚 ∈ dom ran 𝑚 ∈ ω))
234233adantr 474 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ dom ran 𝑚 ∈ ω))
235 id 22 . . . . . . . . . . 11 (𝑚 ∈ ω → 𝑚 ∈ ω)
236 fnfvelrn 6610 . . . . . . . . . . 11 (( Fn ω ∧ 𝑚 ∈ ω) → (𝑚) ∈ ran )
237146, 235, 236syl2anr 590 . . . . . . . . . 10 ((𝑚 ∈ ω ∧ :ω⟶𝑆) → (𝑚) ∈ ran )
238237adantrr 708 . . . . . . . . 9 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → (𝑚) ∈ ran )
239128simpld 490 . . . . . . . . 9 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → 𝑚 ∈ dom (𝑚))
240 dmeq 5560 . . . . . . . . . 10 (𝑢 = (𝑚) → dom 𝑢 = dom (𝑚))
241240eliuni 4748 . . . . . . . . 9 (((𝑚) ∈ ran 𝑚 ∈ dom (𝑚)) → 𝑚 𝑢 ∈ ran dom 𝑢)
242238, 239, 241syl2anc 579 . . . . . . . 8 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → 𝑚 𝑢 ∈ ran dom 𝑢)
243 dmuni 5570 . . . . . . . 8 dom ran = 𝑢 ∈ ran dom 𝑢
244242, 243syl6eleqr 2917 . . . . . . 7 ((𝑚 ∈ ω ∧ (:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))) → 𝑚 ∈ dom ran )
245244expcom 404 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → 𝑚 ∈ dom ran ))
246234, 245impbid 204 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ dom ran 𝑚 ∈ ω))
247246eqrdv 2823 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → dom ran = ω)
248 rnuni 5789 . . . . . 6 ran ran = 𝑠 ∈ ran ran 𝑠
249 frn 6288 . . . . . . . . . . . . . 14 (𝑠:suc 𝑛𝐴 → ran 𝑠𝐴)
2502493ad2ant1 1167 . . . . . . . . . . . . 13 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → ran 𝑠𝐴)
251250rexlimivw 3238 . . . . . . . . . . . 12 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → ran 𝑠𝐴)
252251ss2abi 3901 . . . . . . . . . . 11 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ ran 𝑠𝐴}
25328, 252eqsstri 3860 . . . . . . . . . 10 𝑆 ⊆ {𝑠 ∣ ran 𝑠𝐴}
254133, 253syl6ss 3839 . . . . . . . . 9 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ ran 𝑠𝐴})
255 ssel 3821 . . . . . . . . . 10 (ran ⊆ {𝑠 ∣ ran 𝑠𝐴} → (𝑠 ∈ ran 𝑠 ∈ {𝑠 ∣ ran 𝑠𝐴}))
256 abid 2813 . . . . . . . . . 10 (𝑠 ∈ {𝑠 ∣ ran 𝑠𝐴} ↔ ran 𝑠𝐴)
257255, 256syl6ib 243 . . . . . . . . 9 (ran ⊆ {𝑠 ∣ ran 𝑠𝐴} → (𝑠 ∈ ran → ran 𝑠𝐴))
258254, 257syl 17 . . . . . . . 8 (:ω⟶𝑆 → (𝑠 ∈ ran → ran 𝑠𝐴))
259258ralrimiv 3174 . . . . . . 7 (:ω⟶𝑆 → ∀𝑠 ∈ ran ran 𝑠𝐴)
260 iunss 4783 . . . . . . 7 ( 𝑠 ∈ ran ran 𝑠𝐴 ↔ ∀𝑠 ∈ ran ran 𝑠𝐴)
261259, 260sylibr 226 . . . . . 6 (:ω⟶𝑆 𝑠 ∈ ran ran 𝑠𝐴)
262248, 261syl5eqss 3874 . . . . 5 (:ω⟶𝑆 → ran ran 𝐴)
263262adantr 474 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ran ran 𝐴)
264 df-fn 6130 . . . . 5 ( ran Fn ω ↔ (Fun ran ∧ dom ran = ω))
265 df-f 6131 . . . . . 6 ( ran :ω⟶𝐴 ↔ ( ran Fn ω ∧ ran ran 𝐴))
266265biimpri 220 . . . . 5 (( ran Fn ω ∧ ran ran 𝐴) → ran :ω⟶𝐴)
267264, 266sylanbr 577 . . . 4 (((Fun ran ∧ dom ran = ω) ∧ ran ran 𝐴) → ran :ω⟶𝐴)
268209, 247, 263, 267syl21anc 871 . . 3 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ran :ω⟶𝐴)
269 fnfvelrn 6610 . . . . . . . 8 (( Fn ω ∧ ∅ ∈ ω) → (‘∅) ∈ ran )
270146, 25, 269sylancl 580 . . . . . . 7 (:ω⟶𝑆 → (‘∅) ∈ ran )
271270adantr 474 . . . . . 6 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (‘∅) ∈ ran )
272 elssuni 4691 . . . . . 6 ((‘∅) ∈ ran → (‘∅) ⊆ ran )
273271, 272syl 17 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (‘∅) ⊆ ran )
27454adantr 474 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∅ ∈ dom (‘∅))
275 funssfv 6458 . . . . 5 ((Fun ran ∧ (‘∅) ⊆ ran ∧ ∅ ∈ dom (‘∅)) → ( ran ‘∅) = ((‘∅)‘∅))
276209, 273, 274, 275syl3anc 1494 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ( ran ‘∅) = ((‘∅)‘∅))
277 simp2 1171 . . . . . . . . . . 11 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑠‘∅) = 𝐶)
278277rexlimivw 3238 . . . . . . . . . 10 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑠‘∅) = 𝐶)
279278ss2abi 3901 . . . . . . . . 9 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶}
28028, 279eqsstri 3860 . . . . . . . 8 𝑆 ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶}
281133, 280syl6ss 3839 . . . . . . 7 (:ω⟶𝑆 → ran ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶})
282 ssel 3821 . . . . . . . 8 (ran ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶} → ((‘∅) ∈ ran → (‘∅) ∈ {𝑠 ∣ (𝑠‘∅) = 𝐶}))
283 fveq1 6436 . . . . . . . . . 10 (𝑠 = (‘∅) → (𝑠‘∅) = ((‘∅)‘∅))
284283eqeq1d 2827 . . . . . . . . 9 (𝑠 = (‘∅) → ((𝑠‘∅) = 𝐶 ↔ ((‘∅)‘∅) = 𝐶))
28546, 284elab 3571 . . . . . . . 8 ((‘∅) ∈ {𝑠 ∣ (𝑠‘∅) = 𝐶} ↔ ((‘∅)‘∅) = 𝐶)
286282, 285syl6ib 243 . . . . . . 7 (ran ⊆ {𝑠 ∣ (𝑠‘∅) = 𝐶} → ((‘∅) ∈ ran → ((‘∅)‘∅) = 𝐶))
287281, 286syl 17 . . . . . 6 (:ω⟶𝑆 → ((‘∅) ∈ ran → ((‘∅)‘∅) = 𝐶))
288287adantr 474 . . . . 5 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ((‘∅) ∈ ran → ((‘∅)‘∅) = 𝐶))
289271, 288mpd 15 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ((‘∅)‘∅) = 𝐶)
290276, 289eqtrd 2861 . . 3 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ( ran ‘∅) = 𝐶)
291 nfv 2013 . . . . 5 𝑘 :ω⟶𝑆
292 nfra1 3150 . . . . 5 𝑘𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))
293291, 292nfan 2002 . . . 4 𝑘(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘)))
294133ad2antrr 717 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ran 𝑆)
295 peano2 7352 . . . . . . . . 9 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
296 fnfvelrn 6610 . . . . . . . . 9 (( Fn ω ∧ suc 𝑘 ∈ ω) → (‘suc 𝑘) ∈ ran )
297146, 295, 296syl2an 589 . . . . . . . 8 ((:ω⟶𝑆𝑘 ∈ ω) → (‘suc 𝑘) ∈ ran )
298297adantlr 706 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (‘suc 𝑘) ∈ ran )
299239expcom 404 . . . . . . . . 9 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑚 ∈ ω → 𝑚 ∈ dom (𝑚)))
300299ralrimiv 3174 . . . . . . . 8 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∀𝑚 ∈ ω 𝑚 ∈ dom (𝑚))
301 id 22 . . . . . . . . . . 11 (𝑚 = suc 𝑘𝑚 = suc 𝑘)
302 fveq2 6437 . . . . . . . . . . . 12 (𝑚 = suc 𝑘 → (𝑚) = (‘suc 𝑘))
303302dmeqd 5562 . . . . . . . . . . 11 (𝑚 = suc 𝑘 → dom (𝑚) = dom (‘suc 𝑘))
304301, 303eleq12d 2900 . . . . . . . . . 10 (𝑚 = suc 𝑘 → (𝑚 ∈ dom (𝑚) ↔ suc 𝑘 ∈ dom (‘suc 𝑘)))
305304rspcv 3522 . . . . . . . . 9 (suc 𝑘 ∈ ω → (∀𝑚 ∈ ω 𝑚 ∈ dom (𝑚) → suc 𝑘 ∈ dom (‘suc 𝑘)))
306295, 305syl 17 . . . . . . . 8 (𝑘 ∈ ω → (∀𝑚 ∈ ω 𝑚 ∈ dom (𝑚) → suc 𝑘 ∈ dom (‘suc 𝑘)))
307300, 306mpan9 502 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → suc 𝑘 ∈ dom (‘suc 𝑘))
308 eleq2 2895 . . . . . . . . . . . . . . . . . . . . 21 (dom 𝑠 = suc 𝑛 → (suc 𝑘 ∈ dom 𝑠 ↔ suc 𝑘 ∈ suc 𝑛))
309308biimpa 470 . . . . . . . . . . . . . . . . . . . 20 ((dom 𝑠 = suc 𝑛 ∧ suc 𝑘 ∈ dom 𝑠) → suc 𝑘 ∈ suc 𝑛)
31029, 309sylan 575 . . . . . . . . . . . . . . . . . . 19 ((𝑠:suc 𝑛𝐴 ∧ suc 𝑘 ∈ dom 𝑠) → suc 𝑘 ∈ suc 𝑛)
311 ordsucelsuc 7288 . . . . . . . . . . . . . . . . . . . . . . 23 (Ord 𝑛 → (𝑘𝑛 ↔ suc 𝑘 ∈ suc 𝑛))
31230, 311syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ ω → (𝑘𝑛 ↔ suc 𝑘 ∈ suc 𝑛))
313312biimprd 240 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ω → (suc 𝑘 ∈ suc 𝑛𝑘𝑛))
314 rsp 3138 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑘𝑛 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
315313, 314syl9r 78 . . . . . . . . . . . . . . . . . . . 20 (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑛 ∈ ω → (suc 𝑘 ∈ suc 𝑛 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
316315com13 88 . . . . . . . . . . . . . . . . . . 19 (suc 𝑘 ∈ suc 𝑛 → (𝑛 ∈ ω → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
317310, 316syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑠:suc 𝑛𝐴 ∧ suc 𝑘 ∈ dom 𝑠) → (𝑛 ∈ ω → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
318317ex 403 . . . . . . . . . . . . . . . . 17 (𝑠:suc 𝑛𝐴 → (suc 𝑘 ∈ dom 𝑠 → (𝑛 ∈ ω → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))))
319318com24 95 . . . . . . . . . . . . . . . 16 (𝑠:suc 𝑛𝐴 → (∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) → (𝑛 ∈ ω → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))))
320319imp 397 . . . . . . . . . . . . . . 15 ((𝑠:suc 𝑛𝐴 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑛 ∈ ω → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
3213203adant2 1165 . . . . . . . . . . . . . 14 ((𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (𝑛 ∈ ω → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))))
322321impcom 398 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))) → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
323322rexlimiva 3237 . . . . . . . . . . . 12 (∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) → (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))))
324323ss2abi 3901 . . . . . . . . . . 11 {𝑠 ∣ ∃𝑛 ∈ ω (𝑠:suc 𝑛𝐴 ∧ (𝑠‘∅) = 𝐶 ∧ ∀𝑘𝑛 (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
32528, 324eqsstri 3860 . . . . . . . . . 10 𝑆 ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}
326 sstr 3835 . . . . . . . . . 10 ((ran 𝑆𝑆 ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}) → ran ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
327325, 326mpan2 682 . . . . . . . . 9 (ran 𝑆 → ran ⊆ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))})
328327sseld 3826 . . . . . . . 8 (ran 𝑆 → ((‘suc 𝑘) ∈ ran → (‘suc 𝑘) ∈ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))}))
329 fvex 6450 . . . . . . . . 9 (‘suc 𝑘) ∈ V
330 dmeq 5560 . . . . . . . . . . 11 (𝑠 = (‘suc 𝑘) → dom 𝑠 = dom (‘suc 𝑘))
331330eleq2d 2892 . . . . . . . . . 10 (𝑠 = (‘suc 𝑘) → (suc 𝑘 ∈ dom 𝑠 ↔ suc 𝑘 ∈ dom (‘suc 𝑘)))
332 fveq1 6436 . . . . . . . . . . 11 (𝑠 = (‘suc 𝑘) → (𝑠‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
333 fveq1 6436 . . . . . . . . . . . 12 (𝑠 = (‘suc 𝑘) → (𝑠𝑘) = ((‘suc 𝑘)‘𝑘))
334333fveq2d 6441 . . . . . . . . . . 11 (𝑠 = (‘suc 𝑘) → (𝐹‘(𝑠𝑘)) = (𝐹‘((‘suc 𝑘)‘𝑘)))
335332, 334eleq12d 2900 . . . . . . . . . 10 (𝑠 = (‘suc 𝑘) → ((𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)) ↔ ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
336331, 335imbi12d 336 . . . . . . . . 9 (𝑠 = (‘suc 𝑘) → ((suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘))) ↔ (suc 𝑘 ∈ dom (‘suc 𝑘) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘)))))
337329, 336elab 3571 . . . . . . . 8 ((‘suc 𝑘) ∈ {𝑠 ∣ (suc 𝑘 ∈ dom 𝑠 → (𝑠‘suc 𝑘) ∈ (𝐹‘(𝑠𝑘)))} ↔ (suc 𝑘 ∈ dom (‘suc 𝑘) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
338328, 337syl6ib 243 . . . . . . 7 (ran 𝑆 → ((‘suc 𝑘) ∈ ran → (suc 𝑘 ∈ dom (‘suc 𝑘) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘)))))
339294, 298, 307, 338syl3c 66 . . . . . 6 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘)))
340209adantr 474 . . . . . . . 8 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → Fun ran )
341 elssuni 4691 . . . . . . . . . 10 ((‘suc 𝑘) ∈ ran → (‘suc 𝑘) ⊆ ran )
342297, 341syl 17 . . . . . . . . 9 ((:ω⟶𝑆𝑘 ∈ ω) → (‘suc 𝑘) ⊆ ran )
343342adantlr 706 . . . . . . . 8 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (‘suc 𝑘) ⊆ ran )
344 funssfv 6458 . . . . . . . 8 ((Fun ran ∧ (‘suc 𝑘) ⊆ ran ∧ suc 𝑘 ∈ dom (‘suc 𝑘)) → ( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
345340, 343, 307, 344syl3anc 1494 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
346215sseld 3826 . . . . . . . . . . . . . . 15 (:ω⟶𝑆 → ((‘suc 𝑘) ∈ ran → (‘suc 𝑘) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)}))
347330eleq2d 2892 . . . . . . . . . . . . . . . . 17 (𝑠 = (‘suc 𝑘) → (∅ ∈ dom 𝑠 ↔ ∅ ∈ dom (‘suc 𝑘)))
348330eleq1d 2891 . . . . . . . . . . . . . . . . 17 (𝑠 = (‘suc 𝑘) → (dom 𝑠 ∈ ω ↔ dom (‘suc 𝑘) ∈ ω))
349347, 348anbi12d 624 . . . . . . . . . . . . . . . 16 (𝑠 = (‘suc 𝑘) → ((∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω) ↔ (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω)))
350329, 349elab 3571 . . . . . . . . . . . . . . 15 ((‘suc 𝑘) ∈ {𝑠 ∣ (∅ ∈ dom 𝑠 ∧ dom 𝑠 ∈ ω)} ↔ (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω))
351346, 350syl6ib 243 . . . . . . . . . . . . . 14 (:ω⟶𝑆 → ((‘suc 𝑘) ∈ ran → (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω)))
352351adantr 474 . . . . . . . . . . . . 13 ((:ω⟶𝑆𝑘 ∈ ω) → ((‘suc 𝑘) ∈ ran → (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω)))
353297, 352mpd 15 . . . . . . . . . . . 12 ((:ω⟶𝑆𝑘 ∈ ω) → (∅ ∈ dom (‘suc 𝑘) ∧ dom (‘suc 𝑘) ∈ ω))
354353simprd 491 . . . . . . . . . . 11 ((:ω⟶𝑆𝑘 ∈ ω) → dom (‘suc 𝑘) ∈ ω)
355 nnord 7339 . . . . . . . . . . 11 (dom (‘suc 𝑘) ∈ ω → Ord dom (‘suc 𝑘))
356 ordtr 5981 . . . . . . . . . . 11 (Ord dom (‘suc 𝑘) → Tr dom (‘suc 𝑘))
357 trsuc 6051 . . . . . . . . . . . 12 ((Tr dom (‘suc 𝑘) ∧ suc 𝑘 ∈ dom (‘suc 𝑘)) → 𝑘 ∈ dom (‘suc 𝑘))
358357ex 403 . . . . . . . . . . 11 (Tr dom (‘suc 𝑘) → (suc 𝑘 ∈ dom (‘suc 𝑘) → 𝑘 ∈ dom (‘suc 𝑘)))
359354, 355, 356, 3584syl 19 . . . . . . . . . 10 ((:ω⟶𝑆𝑘 ∈ ω) → (suc 𝑘 ∈ dom (‘suc 𝑘) → 𝑘 ∈ dom (‘suc 𝑘)))
360359adantlr 706 . . . . . . . . 9 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (suc 𝑘 ∈ dom (‘suc 𝑘) → 𝑘 ∈ dom (‘suc 𝑘)))
361307, 360mpd 15 . . . . . . . 8 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → 𝑘 ∈ dom (‘suc 𝑘))
362 funssfv 6458 . . . . . . . 8 ((Fun ran ∧ (‘suc 𝑘) ⊆ ran 𝑘 ∈ dom (‘suc 𝑘)) → ( ran 𝑘) = ((‘suc 𝑘)‘𝑘))
363340, 343, 361, 362syl3anc 1494 . . . . . . 7 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ( ran 𝑘) = ((‘suc 𝑘)‘𝑘))
364 simpl 476 . . . . . . . 8 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → ( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘))
365 simpr 479 . . . . . . . . 9 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → ( ran 𝑘) = ((‘suc 𝑘)‘𝑘))
366365fveq2d 6441 . . . . . . . 8 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → (𝐹‘( ran 𝑘)) = (𝐹‘((‘suc 𝑘)‘𝑘)))
367364, 366eleq12d 2900 . . . . . . 7 ((( ran ‘suc 𝑘) = ((‘suc 𝑘)‘suc 𝑘) ∧ ( ran 𝑘) = ((‘suc 𝑘)‘𝑘)) → (( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)) ↔ ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
368345, 363, 367syl2anc 579 . . . . . 6 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → (( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)) ↔ ((‘suc 𝑘)‘suc 𝑘) ∈ (𝐹‘((‘suc 𝑘)‘𝑘))))
369339, 368mpbird 249 . . . . 5 (((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) ∧ 𝑘 ∈ ω) → ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)))
370369ex 403 . . . 4 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → (𝑘 ∈ ω → ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))))
371293, 370ralrimi 3166 . . 3 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)))
372 vex 3417 . . . . . 6 ∈ V
373372rnex 7367 . . . . 5 ran ∈ V
374373uniex 7218 . . . 4 ran ∈ V
375 feq1 6263 . . . . 5 (𝑔 = ran → (𝑔:ω⟶𝐴 ran :ω⟶𝐴))
376 fveq1 6436 . . . . . 6 (𝑔 = ran → (𝑔‘∅) = ( ran ‘∅))
377376eqeq1d 2827 . . . . 5 (𝑔 = ran → ((𝑔‘∅) = 𝐶 ↔ ( ran ‘∅) = 𝐶))
378 fveq1 6436 . . . . . . 7 (𝑔 = ran → (𝑔‘suc 𝑘) = ( ran ‘suc 𝑘))
379 fveq1 6436 . . . . . . . 8 (𝑔 = ran → (𝑔𝑘) = ( ran 𝑘))
380379fveq2d 6441 . . . . . . 7 (𝑔 = ran → (𝐹‘(𝑔𝑘)) = (𝐹‘( ran 𝑘)))
381378, 380eleq12d 2900 . . . . . 6 (𝑔 = ran → ((𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘)) ↔ ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))))
382381ralbidv 3195 . . . . 5 (𝑔 = ran → (∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘)) ↔ ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))))
383375, 377, 3823anbi123d 1564 . . . 4 (𝑔 = ran → ((𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))) ↔ ( ran :ω⟶𝐴 ∧ ( ran ‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘)))))
384374, 383spcev 3517 . . 3 (( ran :ω⟶𝐴 ∧ ( ran ‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω ( ran ‘suc 𝑘) ∈ (𝐹‘( ran 𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
385268, 290, 371, 384syl3anc 1494 . 2 ((:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
386385exlimiv 2029 1 (∃(:ω⟶𝑆 ∧ ∀𝑘 ∈ ω (‘suc 𝑘) ∈ (𝐺‘(𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ (𝑔‘∅) = 𝐶 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wo 878  w3o 1110  w3a 1111   = wceq 1656  wex 1878  wcel 2164  {cab 2811  wral 3117  wrex 3118  {crab 3121  Vcvv 3414  wss 3798  c0 4146  cop 4405   cuni 4660   ciun 4742  cmpt 4954  Tr wtr 4977  dom cdm 5346  ran crn 5347  cres 5348  Ord word 5966  suc csuc 5969  Fun wfun 6121   Fn wfn 6122  wf 6123  cfv 6127  ωcom 7331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-dc 9590
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-om 7332  df-1o 7831
This theorem is referenced by:  axdc3lem4  9597
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