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Theorem bnj594 31430
Description: Technical lemma for bnj852 31439. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj594.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj594.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj594.3 (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))
bnj594.7 𝐷 = (ω ∖ {∅})
bnj594.9 (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj594.10 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj594.11 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
bnj594.15 (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
bnj594.16 ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))
bnj594.17 (𝜏 ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]𝜃))
Assertion
Ref Expression
bnj594 ((𝑗𝑛𝜏) → 𝜃)
Distinct variable groups:   𝐴,𝑖,𝑘   𝐷,𝑘   𝑅,𝑖,𝑘   𝜒,𝑘   𝑘,𝜒′   𝑓,𝑖,𝑘,𝑦   𝑔,𝑖,𝑘,𝑦   𝑖,𝑛,𝑘   𝑗,𝑘
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜓(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜒(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)   𝜃(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜏(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝐴(𝑥,𝑦,𝑓,𝑔,𝑗,𝑛)   𝐷(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)   𝑅(𝑥,𝑦,𝑓,𝑔,𝑗,𝑛)   𝜑′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑘,𝑛)   𝜒′(𝑥,𝑦,𝑓,𝑔,𝑖,𝑗,𝑛)

Proof of Theorem bnj594
StepHypRef Expression
1 bnj594.3 . . . . . . . . 9 (𝜒 ↔ (𝑓 Fn 𝑛𝜑𝜓))
21simp2bi 1176 . . . . . . . 8 (𝜒𝜑)
3 bnj594.1 . . . . . . . 8 (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
42, 3sylib 209 . . . . . . 7 (𝜒 → (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5 bnj594.11 . . . . . . . . 9 (𝜒′ ↔ (𝑔 Fn 𝑛𝜑′𝜓′))
65simp2bi 1176 . . . . . . . 8 (𝜒′𝜑′)
7 bnj594.9 . . . . . . . 8 (𝜑′ ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
86, 7sylib 209 . . . . . . 7 (𝜒′ → (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
9 eqtr3 2786 . . . . . . 7 (((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ∧ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)) → (𝑓‘∅) = (𝑔‘∅))
104, 8, 9syl2an 589 . . . . . 6 ((𝜒𝜒′) → (𝑓‘∅) = (𝑔‘∅))
11103adant1 1160 . . . . 5 ((𝑛𝐷𝜒𝜒′) → (𝑓‘∅) = (𝑔‘∅))
12 fveq2 6375 . . . . . 6 (𝑗 = ∅ → (𝑓𝑗) = (𝑓‘∅))
13 fveq2 6375 . . . . . 6 (𝑗 = ∅ → (𝑔𝑗) = (𝑔‘∅))
1412, 13eqeq12d 2780 . . . . 5 (𝑗 = ∅ → ((𝑓𝑗) = (𝑔𝑗) ↔ (𝑓‘∅) = (𝑔‘∅)))
1511, 14syl5ibr 237 . . . 4 (𝑗 = ∅ → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
16 bnj594.15 . . . 4 (𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
1715, 16sylibr 225 . . 3 (𝑗 = ∅ → 𝜃)
1817a1d 25 . 2 (𝑗 = ∅ → ((𝑗𝑛𝜏) → 𝜃))
19 bnj253 31221 . . . . . 6 ((𝑛𝐷𝑛𝐷𝜒𝜒′) ↔ ((𝑛𝐷𝑛𝐷) ∧ 𝜒𝜒′))
20 bnj252 31220 . . . . . 6 ((𝑛𝐷𝑛𝐷𝜒𝜒′) ↔ (𝑛𝐷 ∧ (𝑛𝐷𝜒𝜒′)))
21 anidm 560 . . . . . . 7 ((𝑛𝐷𝑛𝐷) ↔ 𝑛𝐷)
22213anbi1i 1196 . . . . . 6 (((𝑛𝐷𝑛𝐷) ∧ 𝜒𝜒′) ↔ (𝑛𝐷𝜒𝜒′))
2319, 20, 223bitr3i 292 . . . . 5 ((𝑛𝐷 ∧ (𝑛𝐷𝜒𝜒′)) ↔ (𝑛𝐷𝜒𝜒′))
24 df-bnj17 31204 . . . . . . . . . 10 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) ↔ ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷) ∧ 𝜏))
25 bnj594.17 . . . . . . . . . . . 12 (𝜏 ↔ ∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]𝜃))
2625bnj1095 31300 . . . . . . . . . . 11 (𝜏 → ∀𝑘𝜏)
2726bnj1352 31346 . . . . . . . . . 10 (((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷) ∧ 𝜏) → ∀𝑘((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷) ∧ 𝜏))
2824, 27hbxfrbi 1919 . . . . . . . . 9 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) → ∀𝑘(𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏))
29 bnj170 31215 . . . . . . . . . . . 12 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷) ↔ ((𝑗𝑛𝑛𝐷) ∧ 𝑗 ≠ ∅))
30 bnj594.7 . . . . . . . . . . . . . . 15 𝐷 = (ω ∖ {∅})
3130bnj923 31286 . . . . . . . . . . . . . 14 (𝑛𝐷𝑛 ∈ ω)
32 elnn 7273 . . . . . . . . . . . . . 14 ((𝑗𝑛𝑛 ∈ ω) → 𝑗 ∈ ω)
3331, 32sylan2 586 . . . . . . . . . . . . 13 ((𝑗𝑛𝑛𝐷) → 𝑗 ∈ ω)
3433anim1i 608 . . . . . . . . . . . 12 (((𝑗𝑛𝑛𝐷) ∧ 𝑗 ≠ ∅) → (𝑗 ∈ ω ∧ 𝑗 ≠ ∅))
3529, 34sylbi 208 . . . . . . . . . . 11 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷) → (𝑗 ∈ ω ∧ 𝑗 ≠ ∅))
36 nnsuc 7280 . . . . . . . . . . 11 ((𝑗 ∈ ω ∧ 𝑗 ≠ ∅) → ∃𝑘 ∈ ω 𝑗 = suc 𝑘)
37 rexex 3148 . . . . . . . . . . 11 (∃𝑘 ∈ ω 𝑗 = suc 𝑘 → ∃𝑘 𝑗 = suc 𝑘)
3835, 36, 373syl 18 . . . . . . . . . 10 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷) → ∃𝑘 𝑗 = suc 𝑘)
3938bnj721 31275 . . . . . . . . 9 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) → ∃𝑘 𝑗 = suc 𝑘)
4028, 39bnj596 31264 . . . . . . . 8 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) → ∃𝑘((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) ∧ 𝑗 = suc 𝑘))
41 bnj667 31270 . . . . . . . . . . 11 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) → (𝑗𝑛𝑛𝐷𝜏))
4241anim1i 608 . . . . . . . . . 10 (((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) ∧ 𝑗 = suc 𝑘) → ((𝑗𝑛𝑛𝐷𝜏) ∧ 𝑗 = suc 𝑘))
43 bnj258 31225 . . . . . . . . . 10 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) ↔ ((𝑗𝑛𝑛𝐷𝜏) ∧ 𝑗 = suc 𝑘))
4442, 43sylibr 225 . . . . . . . . 9 (((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) ∧ 𝑗 = suc 𝑘) → (𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏))
45 df-bnj17 31204 . . . . . . . . . . . . . . 15 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) ↔ ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ 𝜏))
46 bnj219 31250 . . . . . . . . . . . . . . . . . 18 (𝑗 = suc 𝑘𝑘 E 𝑗)
47463ad2ant3 1165 . . . . . . . . . . . . . . . . 17 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) → 𝑘 E 𝑗)
4847adantr 472 . . . . . . . . . . . . . . . 16 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ 𝜏) → 𝑘 E 𝑗)
49 vex 3353 . . . . . . . . . . . . . . . . . . 19 𝑘 ∈ V
5049bnj216 31249 . . . . . . . . . . . . . . . . . 18 (𝑗 = suc 𝑘𝑘𝑗)
51 df-3an 1109 . . . . . . . . . . . . . . . . . . . 20 ((𝑘𝑗𝑗𝑛𝑛𝐷) ↔ ((𝑘𝑗𝑗𝑛) ∧ 𝑛𝐷))
52 3anrot 1122 . . . . . . . . . . . . . . . . . . . 20 ((𝑘𝑗𝑗𝑛𝑛𝐷) ↔ (𝑗𝑛𝑛𝐷𝑘𝑗))
53 ancom 452 . . . . . . . . . . . . . . . . . . . 20 (((𝑘𝑗𝑗𝑛) ∧ 𝑛𝐷) ↔ (𝑛𝐷 ∧ (𝑘𝑗𝑗𝑛)))
5451, 52, 533bitr3i 292 . . . . . . . . . . . . . . . . . . 19 ((𝑗𝑛𝑛𝐷𝑘𝑗) ↔ (𝑛𝐷 ∧ (𝑘𝑗𝑗𝑛)))
55 eldifi 3894 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (ω ∖ {∅}) → 𝑛 ∈ ω)
5655, 30eleq2s 2862 . . . . . . . . . . . . . . . . . . . . 21 (𝑛𝐷𝑛 ∈ ω)
57 nnord 7271 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ω → Ord 𝑛)
58 ordtr1 5951 . . . . . . . . . . . . . . . . . . . . 21 (Ord 𝑛 → ((𝑘𝑗𝑗𝑛) → 𝑘𝑛))
5956, 57, 583syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝑛𝐷 → ((𝑘𝑗𝑗𝑛) → 𝑘𝑛))
6059imp 395 . . . . . . . . . . . . . . . . . . 19 ((𝑛𝐷 ∧ (𝑘𝑗𝑗𝑛)) → 𝑘𝑛)
6154, 60sylbi 208 . . . . . . . . . . . . . . . . . 18 ((𝑗𝑛𝑛𝐷𝑘𝑗) → 𝑘𝑛)
6250, 61syl3an3 1205 . . . . . . . . . . . . . . . . 17 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) → 𝑘𝑛)
63 rsp 3076 . . . . . . . . . . . . . . . . . 18 (∀𝑘𝑛 (𝑘 E 𝑗[𝑘 / 𝑗]𝜃) → (𝑘𝑛 → (𝑘 E 𝑗[𝑘 / 𝑗]𝜃)))
6425, 63sylbi 208 . . . . . . . . . . . . . . . . 17 (𝜏 → (𝑘𝑛 → (𝑘 E 𝑗[𝑘 / 𝑗]𝜃)))
6562, 64mpan9 502 . . . . . . . . . . . . . . . 16 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ 𝜏) → (𝑘 E 𝑗[𝑘 / 𝑗]𝜃))
6648, 65mpd 15 . . . . . . . . . . . . . . 15 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ 𝜏) → [𝑘 / 𝑗]𝜃)
6745, 66sylbi 208 . . . . . . . . . . . . . 14 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) → [𝑘 / 𝑗]𝜃)
6867anim1i 608 . . . . . . . . . . . . 13 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) ∧ (𝑛𝐷𝜒𝜒′)) → ([𝑘 / 𝑗]𝜃 ∧ (𝑛𝐷𝜒𝜒′)))
69 bnj252 31220 . . . . . . . . . . . . 13 (([𝑘 / 𝑗]𝜃𝑛𝐷𝜒𝜒′) ↔ ([𝑘 / 𝑗]𝜃 ∧ (𝑛𝐷𝜒𝜒′)))
7068, 69sylibr 225 . . . . . . . . . . . 12 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) ∧ (𝑛𝐷𝜒𝜒′)) → ([𝑘 / 𝑗]𝜃𝑛𝐷𝜒𝜒′))
71 bnj446 31234 . . . . . . . . . . . . 13 (([𝑘 / 𝑗]𝜃𝑛𝐷𝜒𝜒′) ↔ ((𝑛𝐷𝜒𝜒′) ∧ [𝑘 / 𝑗]𝜃))
72 bnj594.16 . . . . . . . . . . . . . 14 ([𝑘 / 𝑗]𝜃 ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘)))
73 pm3.35 837 . . . . . . . . . . . . . 14 (((𝑛𝐷𝜒𝜒′) ∧ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘))) → (𝑓𝑘) = (𝑔𝑘))
7472, 73sylan2b 587 . . . . . . . . . . . . 13 (((𝑛𝐷𝜒𝜒′) ∧ [𝑘 / 𝑗]𝜃) → (𝑓𝑘) = (𝑔𝑘))
7571, 74sylbi 208 . . . . . . . . . . . 12 (([𝑘 / 𝑗]𝜃𝑛𝐷𝜒𝜒′) → (𝑓𝑘) = (𝑔𝑘))
76 iuneq1 4690 . . . . . . . . . . . 12 ((𝑓𝑘) = (𝑔𝑘) → 𝑦 ∈ (𝑓𝑘) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑔𝑘) pred(𝑦, 𝐴, 𝑅))
7770, 75, 763syl 18 . . . . . . . . . . 11 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) ∧ (𝑛𝐷𝜒𝜒′)) → 𝑦 ∈ (𝑓𝑘) pred(𝑦, 𝐴, 𝑅) = 𝑦 ∈ (𝑔𝑘) pred(𝑦, 𝐴, 𝑅))
78 bnj658 31269 . . . . . . . . . . . . 13 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) → (𝑗𝑛𝑛𝐷𝑗 = suc 𝑘))
791simp3bi 1177 . . . . . . . . . . . . . 14 (𝜒𝜓)
805simp3bi 1177 . . . . . . . . . . . . . 14 (𝜒′𝜓′)
8179, 80bnj240 31216 . . . . . . . . . . . . 13 ((𝑛𝐷𝜒𝜒′) → (𝜓𝜓′))
8278, 81anim12i 606 . . . . . . . . . . . 12 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) ∧ (𝑛𝐷𝜒𝜒′)) → ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ (𝜓𝜓′)))
83 simpl 474 . . . . . . . . . . . . 13 ((𝜓𝜓′) → 𝜓)
8483anim2i 610 . . . . . . . . . . . 12 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ (𝜓𝜓′)) → ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ 𝜓))
85 simp3 1168 . . . . . . . . . . . . . 14 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) → 𝑗 = suc 𝑘)
8685anim1i 608 . . . . . . . . . . . . 13 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ 𝜓) → (𝑗 = suc 𝑘𝜓))
87 simpl1 1242 . . . . . . . . . . . . . 14 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ (𝑗 = suc 𝑘𝜓)) → 𝑗𝑛)
88 df-3an 1109 . . . . . . . . . . . . . . . . 17 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ↔ ((𝑗𝑛𝑛𝐷) ∧ 𝑗 = suc 𝑘))
89 ancom 452 . . . . . . . . . . . . . . . . 17 (((𝑗𝑛𝑛𝐷) ∧ 𝑗 = suc 𝑘) ↔ (𝑗 = suc 𝑘 ∧ (𝑗𝑛𝑛𝐷)))
9088, 89bitri 266 . . . . . . . . . . . . . . . 16 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ↔ (𝑗 = suc 𝑘 ∧ (𝑗𝑛𝑛𝐷)))
91 elnn 7273 . . . . . . . . . . . . . . . . 17 ((𝑘𝑗𝑗 ∈ ω) → 𝑘 ∈ ω)
9250, 33, 91syl2an 589 . . . . . . . . . . . . . . . 16 ((𝑗 = suc 𝑘 ∧ (𝑗𝑛𝑛𝐷)) → 𝑘 ∈ ω)
9390, 92sylbi 208 . . . . . . . . . . . . . . 15 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) → 𝑘 ∈ ω)
94 bnj594.2 . . . . . . . . . . . . . . . . 17 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
9594bnj589 31427 . . . . . . . . . . . . . . . 16 (𝜓 ↔ ∀𝑘 ∈ ω (suc 𝑘𝑛 → (𝑓‘suc 𝑘) = 𝑦 ∈ (𝑓𝑘) pred(𝑦, 𝐴, 𝑅)))
9695bnj590 31428 . . . . . . . . . . . . . . 15 ((𝑗 = suc 𝑘𝜓) → (𝑘 ∈ ω → (𝑗𝑛 → (𝑓𝑗) = 𝑦 ∈ (𝑓𝑘) pred(𝑦, 𝐴, 𝑅))))
9793, 96mpan9 502 . . . . . . . . . . . . . 14 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ (𝑗 = suc 𝑘𝜓)) → (𝑗𝑛 → (𝑓𝑗) = 𝑦 ∈ (𝑓𝑘) pred(𝑦, 𝐴, 𝑅)))
9887, 97mpd 15 . . . . . . . . . . . . 13 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ (𝑗 = suc 𝑘𝜓)) → (𝑓𝑗) = 𝑦 ∈ (𝑓𝑘) pred(𝑦, 𝐴, 𝑅))
9986, 98syldan 585 . . . . . . . . . . . 12 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ 𝜓) → (𝑓𝑗) = 𝑦 ∈ (𝑓𝑘) pred(𝑦, 𝐴, 𝑅))
10082, 84, 993syl 18 . . . . . . . . . . 11 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = 𝑦 ∈ (𝑓𝑘) pred(𝑦, 𝐴, 𝑅))
101 simpr 477 . . . . . . . . . . . . 13 ((𝜓𝜓′) → 𝜓′)
102101anim2i 610 . . . . . . . . . . . 12 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ (𝜓𝜓′)) → ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ 𝜓′))
10385anim1i 608 . . . . . . . . . . . . 13 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ 𝜓′) → (𝑗 = suc 𝑘𝜓′))
104 simpl1 1242 . . . . . . . . . . . . . 14 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ (𝑗 = suc 𝑘𝜓′)) → 𝑗𝑛)
105 bnj594.10 . . . . . . . . . . . . . . . . 17 (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑔‘suc 𝑖) = 𝑦 ∈ (𝑔𝑖) pred(𝑦, 𝐴, 𝑅)))
106105bnj589 31427 . . . . . . . . . . . . . . . 16 (𝜓′ ↔ ∀𝑘 ∈ ω (suc 𝑘𝑛 → (𝑔‘suc 𝑘) = 𝑦 ∈ (𝑔𝑘) pred(𝑦, 𝐴, 𝑅)))
107106bnj590 31428 . . . . . . . . . . . . . . 15 ((𝑗 = suc 𝑘𝜓′) → (𝑘 ∈ ω → (𝑗𝑛 → (𝑔𝑗) = 𝑦 ∈ (𝑔𝑘) pred(𝑦, 𝐴, 𝑅))))
10893, 107mpan9 502 . . . . . . . . . . . . . 14 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ (𝑗 = suc 𝑘𝜓′)) → (𝑗𝑛 → (𝑔𝑗) = 𝑦 ∈ (𝑔𝑘) pred(𝑦, 𝐴, 𝑅)))
109104, 108mpd 15 . . . . . . . . . . . . 13 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ (𝑗 = suc 𝑘𝜓′)) → (𝑔𝑗) = 𝑦 ∈ (𝑔𝑘) pred(𝑦, 𝐴, 𝑅))
110103, 109syldan 585 . . . . . . . . . . . 12 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘) ∧ 𝜓′) → (𝑔𝑗) = 𝑦 ∈ (𝑔𝑘) pred(𝑦, 𝐴, 𝑅))
11182, 102, 1103syl 18 . . . . . . . . . . 11 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) ∧ (𝑛𝐷𝜒𝜒′)) → (𝑔𝑗) = 𝑦 ∈ (𝑔𝑘) pred(𝑦, 𝐴, 𝑅))
11277, 100, 1113eqtr4d 2809 . . . . . . . . . 10 (((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = (𝑔𝑗))
113112ex 401 . . . . . . . . 9 ((𝑗𝑛𝑛𝐷𝑗 = suc 𝑘𝜏) → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
11444, 113syl 17 . . . . . . . 8 (((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) ∧ 𝑗 = suc 𝑘) → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
11540, 114bnj593 31263 . . . . . . 7 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) → ∃𝑘((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
116 bnj258 31225 . . . . . . 7 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝑛𝐷𝜏) ↔ ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝜏) ∧ 𝑛𝐷))
117 19.9v 2078 . . . . . . 7 (∃𝑘((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)) ↔ ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
118115, 116, 1173imtr3i 282 . . . . . 6 (((𝑗 ≠ ∅ ∧ 𝑗𝑛𝜏) ∧ 𝑛𝐷) → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
119118expimpd 445 . . . . 5 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝜏) → ((𝑛𝐷 ∧ (𝑛𝐷𝜒𝜒′)) → (𝑓𝑗) = (𝑔𝑗)))
12023, 119syl5bir 234 . . . 4 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝜏) → ((𝑛𝐷𝜒𝜒′) → (𝑓𝑗) = (𝑔𝑗)))
121120, 16sylibr 225 . . 3 ((𝑗 ≠ ∅ ∧ 𝑗𝑛𝜏) → 𝜃)
1221213expib 1152 . 2 (𝑗 ≠ ∅ → ((𝑗𝑛𝜏) → 𝜃))
12318, 122pm2.61ine 3020 1 ((𝑗𝑛𝜏) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wne 2937  wral 3055  wrex 3056  [wsbc 3596  cdif 3729  c0 4079  {csn 4334   ciun 4676   class class class wbr 4809   E cep 5189  Ord word 5907  suc csuc 5910   Fn wfn 6063  cfv 6068  ωcom 7263  w-bnj17 31203   predc-bnj14 31205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-tr 4912  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fv 6076  df-om 7264  df-bnj17 31204
This theorem is referenced by:  bnj580  31431
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