MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axdc2lem Structured version   Visualization version   GIF version

Theorem axdc2lem 9462
Description: Lemma for axdc2 9463. We construct a relation 𝑅 based on 𝐹 such that 𝑥𝑅𝑦 iff 𝑦 ∈ (𝐹𝑥), and show that the "function" described by ax-dc 9460 can be restricted so that it is a real function (since the stated properties only show that it is the superset of a function). (Contributed by Mario Carneiro, 25-Jan-2013.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
axdc2lem.1 𝐴 ∈ V
axdc2lem.2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
axdc2lem.3 𝐺 = (𝑥 ∈ ω ↦ (𝑥))
Assertion
Ref Expression
axdc2lem ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
Distinct variable groups:   𝐴,𝑔,   𝑥,𝐴,𝑦,   𝑔,𝐹,   𝑥,𝐹,𝑦   𝑔,𝐺,𝑘   𝑥,𝐺,𝑦,𝑘   𝑅,,𝑘,𝑥
Allowed substitution hints:   𝐴(𝑘)   𝑅(𝑦,𝑔)   𝐹(𝑘)   𝐺()

Proof of Theorem axdc2lem
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 ffvelrn 6520 . . . . . . . . 9 ((𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) ∧ 𝑥𝐴) → (𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}))
2 eldifsni 4466 . . . . . . . . . 10 ((𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}) → (𝐹𝑥) ≠ ∅)
3 n0 4074 . . . . . . . . . 10 ((𝐹𝑥) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑥))
42, 3sylib 208 . . . . . . . . 9 ((𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}) → ∃𝑦 𝑦 ∈ (𝐹𝑥))
51, 4syl 17 . . . . . . . 8 ((𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) ∧ 𝑥𝐴) → ∃𝑦 𝑦 ∈ (𝐹𝑥))
65ralrimiva 3104 . . . . . . 7 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ∀𝑥𝐴𝑦 𝑦 ∈ (𝐹𝑥))
7 rabid2 3257 . . . . . . 7 (𝐴 = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} ↔ ∀𝑥𝐴𝑦 𝑦 ∈ (𝐹𝑥))
86, 7sylibr 224 . . . . . 6 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → 𝐴 = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)})
9 axdc2lem.2 . . . . . . . 8 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
109dmeqi 5480 . . . . . . 7 dom 𝑅 = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
11 19.42v 2030 . . . . . . . . 9 (∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥)))
1211abbii 2877 . . . . . . . 8 {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥))}
13 dmopab 5490 . . . . . . . 8 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑥 ∣ ∃𝑦(𝑥𝐴𝑦 ∈ (𝐹𝑥))}
14 df-rab 3059 . . . . . . . 8 {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)} = {𝑥 ∣ (𝑥𝐴 ∧ ∃𝑦 𝑦 ∈ (𝐹𝑥))}
1512, 13, 143eqtr4i 2792 . . . . . . 7 dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)}
1610, 15eqtri 2782 . . . . . 6 dom 𝑅 = {𝑥𝐴 ∣ ∃𝑦 𝑦 ∈ (𝐹𝑥)}
178, 16syl6reqr 2813 . . . . 5 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → dom 𝑅 = 𝐴)
1817neeq1d 2991 . . . 4 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → (dom 𝑅 ≠ ∅ ↔ 𝐴 ≠ ∅))
1918biimparc 505 . . 3 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → dom 𝑅 ≠ ∅)
20 eldifi 3875 . . . . . . . . . 10 ((𝐹𝑥) ∈ (𝒫 𝐴 ∖ {∅}) → (𝐹𝑥) ∈ 𝒫 𝐴)
21 elelpwi 4315 . . . . . . . . . . 11 ((𝑦 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ 𝒫 𝐴) → 𝑦𝐴)
2221expcom 450 . . . . . . . . . 10 ((𝐹𝑥) ∈ 𝒫 𝐴 → (𝑦 ∈ (𝐹𝑥) → 𝑦𝐴))
231, 20, 223syl 18 . . . . . . . . 9 ((𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝐹𝑥) → 𝑦𝐴))
2423expimpd 630 . . . . . . . 8 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ((𝑥𝐴𝑦 ∈ (𝐹𝑥)) → 𝑦𝐴))
2524exlimdv 2010 . . . . . . 7 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → (∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥)) → 𝑦𝐴))
2625alrimiv 2004 . . . . . 6 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ∀𝑦(∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥)) → 𝑦𝐴))
279rneqi 5507 . . . . . . . . 9 ran 𝑅 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
28 rnopab 5525 . . . . . . . . 9 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))}
2927, 28eqtri 2782 . . . . . . . 8 ran 𝑅 = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))}
3029sseq1i 3770 . . . . . . 7 (ran 𝑅𝐴 ↔ {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ⊆ 𝐴)
31 abss 3812 . . . . . . 7 ({𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥))} ⊆ 𝐴 ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥)) → 𝑦𝐴))
3230, 31bitri 264 . . . . . 6 (ran 𝑅𝐴 ↔ ∀𝑦(∃𝑥(𝑥𝐴𝑦 ∈ (𝐹𝑥)) → 𝑦𝐴))
3326, 32sylibr 224 . . . . 5 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ran 𝑅𝐴)
3433, 17sseqtr4d 3783 . . . 4 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ran 𝑅 ⊆ dom 𝑅)
3534adantl 473 . . 3 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ran 𝑅 ⊆ dom 𝑅)
36 fvrn0 6377 . . . . . . . . . 10 (𝐹𝑥) ∈ (ran 𝐹 ∪ {∅})
37 elssuni 4619 . . . . . . . . . 10 ((𝐹𝑥) ∈ (ran 𝐹 ∪ {∅}) → (𝐹𝑥) ⊆ (ran 𝐹 ∪ {∅}))
3836, 37ax-mp 5 . . . . . . . . 9 (𝐹𝑥) ⊆ (ran 𝐹 ∪ {∅})
3938sseli 3740 . . . . . . . 8 (𝑦 ∈ (𝐹𝑥) → 𝑦 (ran 𝐹 ∪ {∅}))
4039anim2i 594 . . . . . . 7 ((𝑥𝐴𝑦 ∈ (𝐹𝑥)) → (𝑥𝐴𝑦 (ran 𝐹 ∪ {∅})))
4140ssopab2i 5153 . . . . . 6 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 (ran 𝐹 ∪ {∅}))}
42 df-xp 5272 . . . . . 6 (𝐴 × (ran 𝐹 ∪ {∅})) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 (ran 𝐹 ∪ {∅}))}
4341, 9, 423sstr4i 3785 . . . . 5 𝑅 ⊆ (𝐴 × (ran 𝐹 ∪ {∅}))
44 axdc2lem.1 . . . . . 6 𝐴 ∈ V
45 frn 6214 . . . . . . . . . 10 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ran 𝐹 ⊆ (𝒫 𝐴 ∖ {∅}))
4645adantl 473 . . . . . . . . 9 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ran 𝐹 ⊆ (𝒫 𝐴 ∖ {∅}))
4744pwex 4997 . . . . . . . . . . 11 𝒫 𝐴 ∈ V
48 difexg 4960 . . . . . . . . . . 11 (𝒫 𝐴 ∈ V → (𝒫 𝐴 ∖ {∅}) ∈ V)
4947, 48ax-mp 5 . . . . . . . . . 10 (𝒫 𝐴 ∖ {∅}) ∈ V
5049ssex 4954 . . . . . . . . 9 (ran 𝐹 ⊆ (𝒫 𝐴 ∖ {∅}) → ran 𝐹 ∈ V)
5146, 50syl 17 . . . . . . . 8 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ran 𝐹 ∈ V)
52 p0ex 5002 . . . . . . . 8 {∅} ∈ V
53 unexg 7124 . . . . . . . 8 ((ran 𝐹 ∈ V ∧ {∅} ∈ V) → (ran 𝐹 ∪ {∅}) ∈ V)
5451, 52, 53sylancl 697 . . . . . . 7 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → (ran 𝐹 ∪ {∅}) ∈ V)
55 uniexg 7120 . . . . . . 7 ((ran 𝐹 ∪ {∅}) ∈ V → (ran 𝐹 ∪ {∅}) ∈ V)
5654, 55syl 17 . . . . . 6 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → (ran 𝐹 ∪ {∅}) ∈ V)
57 xpexg 7125 . . . . . 6 ((𝐴 ∈ V ∧ (ran 𝐹 ∪ {∅}) ∈ V) → (𝐴 × (ran 𝐹 ∪ {∅})) ∈ V)
5844, 56, 57sylancr 698 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → (𝐴 × (ran 𝐹 ∪ {∅})) ∈ V)
59 ssexg 4956 . . . . 5 ((𝑅 ⊆ (𝐴 × (ran 𝐹 ∪ {∅})) ∧ (𝐴 × (ran 𝐹 ∪ {∅})) ∈ V) → 𝑅 ∈ V)
6043, 58, 59sylancr 698 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → 𝑅 ∈ V)
61 n0 4074 . . . . . . . . 9 (dom 𝑟 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ dom 𝑟)
62 vex 3343 . . . . . . . . . . 11 𝑥 ∈ V
6362eldm 5476 . . . . . . . . . 10 (𝑥 ∈ dom 𝑟 ↔ ∃𝑦 𝑥𝑟𝑦)
6463exbii 1923 . . . . . . . . 9 (∃𝑥 𝑥 ∈ dom 𝑟 ↔ ∃𝑥𝑦 𝑥𝑟𝑦)
6561, 64bitr2i 265 . . . . . . . 8 (∃𝑥𝑦 𝑥𝑟𝑦 ↔ dom 𝑟 ≠ ∅)
66 dmeq 5479 . . . . . . . . 9 (𝑟 = 𝑅 → dom 𝑟 = dom 𝑅)
6766neeq1d 2991 . . . . . . . 8 (𝑟 = 𝑅 → (dom 𝑟 ≠ ∅ ↔ dom 𝑅 ≠ ∅))
6865, 67syl5bb 272 . . . . . . 7 (𝑟 = 𝑅 → (∃𝑥𝑦 𝑥𝑟𝑦 ↔ dom 𝑅 ≠ ∅))
69 rneq 5506 . . . . . . . 8 (𝑟 = 𝑅 → ran 𝑟 = ran 𝑅)
7069, 66sseq12d 3775 . . . . . . 7 (𝑟 = 𝑅 → (ran 𝑟 ⊆ dom 𝑟 ↔ ran 𝑅 ⊆ dom 𝑅))
7168, 70anbi12d 749 . . . . . 6 (𝑟 = 𝑅 → ((∃𝑥𝑦 𝑥𝑟𝑦 ∧ ran 𝑟 ⊆ dom 𝑟) ↔ (dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅)))
72 breq 4806 . . . . . . . 8 (𝑟 = 𝑅 → ((𝑘)𝑟(‘suc 𝑘) ↔ (𝑘)𝑅(‘suc 𝑘)))
7372ralbidv 3124 . . . . . . 7 (𝑟 = 𝑅 → (∀𝑘 ∈ ω (𝑘)𝑟(‘suc 𝑘) ↔ ∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘)))
7473exbidv 1999 . . . . . 6 (𝑟 = 𝑅 → (∃𝑘 ∈ ω (𝑘)𝑟(‘suc 𝑘) ↔ ∃𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘)))
7571, 74imbi12d 333 . . . . 5 (𝑟 = 𝑅 → (((∃𝑥𝑦 𝑥𝑟𝑦 ∧ ran 𝑟 ⊆ dom 𝑟) → ∃𝑘 ∈ ω (𝑘)𝑟(‘suc 𝑘)) ↔ ((dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅) → ∃𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘))))
76 ax-dc 9460 . . . . 5 ((∃𝑥𝑦 𝑥𝑟𝑦 ∧ ran 𝑟 ⊆ dom 𝑟) → ∃𝑘 ∈ ω (𝑘)𝑟(‘suc 𝑘))
7775, 76vtoclg 3406 . . . 4 (𝑅 ∈ V → ((dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅) → ∃𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘)))
7860, 77syl 17 . . 3 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ((dom 𝑅 ≠ ∅ ∧ ran 𝑅 ⊆ dom 𝑅) → ∃𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘)))
7919, 35, 78mp2and 717 . 2 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘))
80 simpr 479 . 2 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}))
81 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑘 = 𝑥 → (𝑘) = (𝑥))
82 suceq 5951 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑥 → suc 𝑘 = suc 𝑥)
8382fveq2d 6356 . . . . . . . . . . . . . . 15 (𝑘 = 𝑥 → (‘suc 𝑘) = (‘suc 𝑥))
8481, 83breq12d 4817 . . . . . . . . . . . . . 14 (𝑘 = 𝑥 → ((𝑘)𝑅(‘suc 𝑘) ↔ (𝑥)𝑅(‘suc 𝑥)))
8584rspccv 3446 . . . . . . . . . . . . 13 (∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) → (𝑥 ∈ ω → (𝑥)𝑅(‘suc 𝑥)))
86 fvex 6362 . . . . . . . . . . . . . 14 (𝑥) ∈ V
87 fvex 6362 . . . . . . . . . . . . . 14 (‘suc 𝑥) ∈ V
8886, 87breldm 5484 . . . . . . . . . . . . 13 ((𝑥)𝑅(‘suc 𝑥) → (𝑥) ∈ dom 𝑅)
8985, 88syl6 35 . . . . . . . . . . . 12 (∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) → (𝑥 ∈ ω → (𝑥) ∈ dom 𝑅))
9089imp 444 . . . . . . . . . . 11 ((∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) ∧ 𝑥 ∈ ω) → (𝑥) ∈ dom 𝑅)
9190adantll 752 . . . . . . . . . 10 (((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘)) ∧ 𝑥 ∈ ω) → (𝑥) ∈ dom 𝑅)
92 eleq2 2828 . . . . . . . . . . 11 (dom 𝑅 = 𝐴 → ((𝑥) ∈ dom 𝑅 ↔ (𝑥) ∈ 𝐴))
9392ad2antrr 764 . . . . . . . . . 10 (((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘)) ∧ 𝑥 ∈ ω) → ((𝑥) ∈ dom 𝑅 ↔ (𝑥) ∈ 𝐴))
9491, 93mpbid 222 . . . . . . . . 9 (((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘)) ∧ 𝑥 ∈ ω) → (𝑥) ∈ 𝐴)
95 axdc2lem.3 . . . . . . . . 9 𝐺 = (𝑥 ∈ ω ↦ (𝑥))
9694, 95fmptd 6548 . . . . . . . 8 ((dom 𝑅 = 𝐴 ∧ ∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘)) → 𝐺:ω⟶𝐴)
9796ex 449 . . . . . . 7 (dom 𝑅 = 𝐴 → (∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) → 𝐺:ω⟶𝐴))
9817, 97syl 17 . . . . . 6 (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → (∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) → 𝐺:ω⟶𝐴))
9998impcom 445 . . . . 5 ((∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → 𝐺:ω⟶𝐴)
100 fveq2 6352 . . . . . . . . . 10 (𝑥 = 𝑘 → (𝑥) = (𝑘))
101 fvex 6362 . . . . . . . . . 10 (𝑘) ∈ V
102100, 95, 101fvmpt 6444 . . . . . . . . 9 (𝑘 ∈ ω → (𝐺𝑘) = (𝑘))
103 peano2 7251 . . . . . . . . . 10 (𝑘 ∈ ω → suc 𝑘 ∈ ω)
104 fvex 6362 . . . . . . . . . 10 (‘suc 𝑘) ∈ V
105 fveq2 6352 . . . . . . . . . . 11 (𝑥 = suc 𝑘 → (𝑥) = (‘suc 𝑘))
106105, 95fvmptg 6442 . . . . . . . . . 10 ((suc 𝑘 ∈ ω ∧ (‘suc 𝑘) ∈ V) → (𝐺‘suc 𝑘) = (‘suc 𝑘))
107103, 104, 106sylancl 697 . . . . . . . . 9 (𝑘 ∈ ω → (𝐺‘suc 𝑘) = (‘suc 𝑘))
108102, 107breq12d 4817 . . . . . . . 8 (𝑘 ∈ ω → ((𝐺𝑘)𝑅(𝐺‘suc 𝑘) ↔ (𝑘)𝑅(‘suc 𝑘)))
109 fvex 6362 . . . . . . . . . 10 (𝐺𝑘) ∈ V
110 fvex 6362 . . . . . . . . . 10 (𝐺‘suc 𝑘) ∈ V
111 eleq1 2827 . . . . . . . . . . 11 (𝑥 = (𝐺𝑘) → (𝑥𝐴 ↔ (𝐺𝑘) ∈ 𝐴))
112 fveq2 6352 . . . . . . . . . . . 12 (𝑥 = (𝐺𝑘) → (𝐹𝑥) = (𝐹‘(𝐺𝑘)))
113112eleq2d 2825 . . . . . . . . . . 11 (𝑥 = (𝐺𝑘) → (𝑦 ∈ (𝐹𝑥) ↔ 𝑦 ∈ (𝐹‘(𝐺𝑘))))
114111, 113anbi12d 749 . . . . . . . . . 10 (𝑥 = (𝐺𝑘) → ((𝑥𝐴𝑦 ∈ (𝐹𝑥)) ↔ ((𝐺𝑘) ∈ 𝐴𝑦 ∈ (𝐹‘(𝐺𝑘)))))
115 eleq1 2827 . . . . . . . . . . 11 (𝑦 = (𝐺‘suc 𝑘) → (𝑦 ∈ (𝐹‘(𝐺𝑘)) ↔ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘))))
116115anbi2d 742 . . . . . . . . . 10 (𝑦 = (𝐺‘suc 𝑘) → (((𝐺𝑘) ∈ 𝐴𝑦 ∈ (𝐹‘(𝐺𝑘))) ↔ ((𝐺𝑘) ∈ 𝐴 ∧ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘)))))
117109, 110, 114, 116, 9brab 5148 . . . . . . . . 9 ((𝐺𝑘)𝑅(𝐺‘suc 𝑘) ↔ ((𝐺𝑘) ∈ 𝐴 ∧ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘))))
118117simprbi 483 . . . . . . . 8 ((𝐺𝑘)𝑅(𝐺‘suc 𝑘) → (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘)))
119108, 118syl6bir 244 . . . . . . 7 (𝑘 ∈ ω → ((𝑘)𝑅(‘suc 𝑘) → (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘))))
120119ralimia 3088 . . . . . 6 (∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) → ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘)))
121120adantr 472 . . . . 5 ((∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘)))
122 fvrn0 6377 . . . . . . . . . 10 (𝑥) ∈ (ran ∪ {∅})
123122rgenw 3062 . . . . . . . . 9 𝑥 ∈ ω (𝑥) ∈ (ran ∪ {∅})
124 eqid 2760 . . . . . . . . . 10 (𝑥 ∈ ω ↦ (𝑥)) = (𝑥 ∈ ω ↦ (𝑥))
125124fmpt 6544 . . . . . . . . 9 (∀𝑥 ∈ ω (𝑥) ∈ (ran ∪ {∅}) ↔ (𝑥 ∈ ω ↦ (𝑥)):ω⟶(ran ∪ {∅}))
126123, 125mpbi 220 . . . . . . . 8 (𝑥 ∈ ω ↦ (𝑥)):ω⟶(ran ∪ {∅})
127 dcomex 9461 . . . . . . . 8 ω ∈ V
128 vex 3343 . . . . . . . . . 10 ∈ V
129128rnex 7265 . . . . . . . . 9 ran ∈ V
130129, 52unex 7121 . . . . . . . 8 (ran ∪ {∅}) ∈ V
131 fex2 7286 . . . . . . . 8 (((𝑥 ∈ ω ↦ (𝑥)):ω⟶(ran ∪ {∅}) ∧ ω ∈ V ∧ (ran ∪ {∅}) ∈ V) → (𝑥 ∈ ω ↦ (𝑥)) ∈ V)
132126, 127, 130, 131mp3an 1573 . . . . . . 7 (𝑥 ∈ ω ↦ (𝑥)) ∈ V
13395, 132eqeltri 2835 . . . . . 6 𝐺 ∈ V
134 feq1 6187 . . . . . . 7 (𝑔 = 𝐺 → (𝑔:ω⟶𝐴𝐺:ω⟶𝐴))
135 fveq1 6351 . . . . . . . . 9 (𝑔 = 𝐺 → (𝑔‘suc 𝑘) = (𝐺‘suc 𝑘))
136 fveq1 6351 . . . . . . . . . 10 (𝑔 = 𝐺 → (𝑔𝑘) = (𝐺𝑘))
137136fveq2d 6356 . . . . . . . . 9 (𝑔 = 𝐺 → (𝐹‘(𝑔𝑘)) = (𝐹‘(𝐺𝑘)))
138135, 137eleq12d 2833 . . . . . . . 8 (𝑔 = 𝐺 → ((𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘)) ↔ (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘))))
139138ralbidv 3124 . . . . . . 7 (𝑔 = 𝐺 → (∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘)) ↔ ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘))))
140134, 139anbi12d 749 . . . . . 6 (𝑔 = 𝐺 → ((𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))) ↔ (𝐺:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘)))))
141133, 140spcev 3440 . . . . 5 ((𝐺:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝐺‘suc 𝑘) ∈ (𝐹‘(𝐺𝑘))) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
14299, 121, 141syl2anc 696 . . . 4 ((∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
143142ex 449 . . 3 (∀𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) → (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘)))))
144143exlimiv 2007 . 2 (∃𝑘 ∈ ω (𝑘)𝑅(‘suc 𝑘) → (𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅}) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘)))))
14579, 80, 144sylc 65 1 ((𝐴 ≠ ∅ ∧ 𝐹:𝐴⟶(𝒫 𝐴 ∖ {∅})) → ∃𝑔(𝑔:ω⟶𝐴 ∧ ∀𝑘 ∈ ω (𝑔‘suc 𝑘) ∈ (𝐹‘(𝑔𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wne 2932  wral 3050  {crab 3054  Vcvv 3340  cdif 3712  cun 3713  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321   cuni 4588   class class class wbr 4804  {copab 4864  cmpt 4881   × cxp 5264  dom cdm 5266  ran crn 5267  suc csuc 5886  wf 6045  cfv 6049  ωcom 7230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-dc 9460
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-om 7231  df-1o 7729
This theorem is referenced by:  axdc2  9463
  Copyright terms: Public domain W3C validator