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Theorem exrecfnlem 36260
Description: Lemma for exrecfn 36261. (Contributed by ML, 30-Mar-2022.)
Hypothesis
Ref Expression
exrecfnlem.1 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
Assertion
Ref Expression
exrecfnlem ((𝐴𝑉 ∧ ∀𝑦 𝐵𝑊) → ∃𝑥(𝐴𝑥 ∧ ∀𝑦𝑥 𝐵𝑥))
Distinct variable groups:   𝑦,𝐴,𝑧,𝑥   𝑥,𝐵,𝑧   𝑥,𝐹   𝑦,𝑊
Allowed substitution hints:   𝐵(𝑦)   𝐹(𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑧)

Proof of Theorem exrecfnlem
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 rdg0g 8427 . . 3 (𝐴𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
2 peano1 7879 . . . 4 ∅ ∈ ω
3 omelon 9641 . . . . 5 ω ∈ On
4 limom 7871 . . . . 5 Lim ω
5 rdglimss 36258 . . . . 5 (((ω ∈ On ∧ Lim ω) ∧ ∅ ∈ ω) → (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω))
63, 4, 5mpanl12 701 . . . 4 (∅ ∈ ω → (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω))
72, 6ax-mp 5 . . 3 (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω)
81, 7eqsstrrdi 4038 . 2 (𝐴𝑉𝐴 ⊆ (rec(𝐹, 𝐴)‘ω))
9 rdglim2a 8433 . . . . . . . 8 ((ω ∈ On ∧ Lim ω) → (rec(𝐹, 𝐴)‘ω) = 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
103, 4, 9mp2an 691 . . . . . . 7 (rec(𝐹, 𝐴)‘ω) = 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢)
1110eleq2i 2826 . . . . . 6 (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ 𝑦 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
12 eliun 5002 . . . . . 6 (𝑦 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢) ↔ ∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢))
1311, 12bitri 275 . . . . 5 (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ ∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢))
14 peano2 7881 . . . . . . . . 9 (𝑢 ∈ ω → suc 𝑢 ∈ ω)
15 nnon 7861 . . . . . . . . . 10 (𝑢 ∈ ω → 𝑢 ∈ On)
16 eqid 2733 . . . . . . . . . . . . 13 (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
1716elrnmpt1 5958 . . . . . . . . . . . 12 ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵𝑊) → 𝐵 ∈ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
18 elun2 4178 . . . . . . . . . . . 12 (𝐵 ∈ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) → 𝐵 ∈ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
1917, 18syl 17 . . . . . . . . . . 11 ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵𝑊) → 𝐵 ∈ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
20 fvex 6905 . . . . . . . . . . . . . 14 (rec(𝐹, 𝐴)‘𝑢) ∈ V
21 exrecfnlem.1 . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
22 nfcv 2904 . . . . . . . . . . . . . . . . . . . . 21 𝑦V
23 nfcv 2904 . . . . . . . . . . . . . . . . . . . . . 22 𝑦𝑧
24 nfmpt1 5257 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦(𝑦𝑧𝐵)
2524nfrn 5952 . . . . . . . . . . . . . . . . . . . . . 22 𝑦ran (𝑦𝑧𝐵)
2623, 25nfun 4166 . . . . . . . . . . . . . . . . . . . . 21 𝑦(𝑧 ∪ ran (𝑦𝑧𝐵))
2722, 26nfmpt 5256 . . . . . . . . . . . . . . . . . . . 20 𝑦(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
2821, 27nfcxfr 2902 . . . . . . . . . . . . . . . . . . 19 𝑦𝐹
29 nfcv 2904 . . . . . . . . . . . . . . . . . . 19 𝑦𝐴
3028, 29nfrdg 8414 . . . . . . . . . . . . . . . . . 18 𝑦rec(𝐹, 𝐴)
31 nfcv 2904 . . . . . . . . . . . . . . . . . 18 𝑦𝑢
3230, 31nffv 6902 . . . . . . . . . . . . . . . . 17 𝑦(rec(𝐹, 𝐴)‘𝑢)
3332mptexgf 7224 . . . . . . . . . . . . . . . 16 ((rec(𝐹, 𝐴)‘𝑢) ∈ V → (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V)
3420, 33ax-mp 5 . . . . . . . . . . . . . . 15 (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V
3534rnex 7903 . . . . . . . . . . . . . 14 ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V
3620, 35unex 7733 . . . . . . . . . . . . 13 ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)) ∈ V
37 nfcv 2904 . . . . . . . . . . . . . 14 𝑧𝐴
38 nfcv 2904 . . . . . . . . . . . . . 14 𝑧𝑢
39 nfmpt1 5257 . . . . . . . . . . . . . . . . . 18 𝑧(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
4021, 39nfcxfr 2902 . . . . . . . . . . . . . . . . 17 𝑧𝐹
4140, 37nfrdg 8414 . . . . . . . . . . . . . . . 16 𝑧rec(𝐹, 𝐴)
4241, 38nffv 6902 . . . . . . . . . . . . . . 15 𝑧(rec(𝐹, 𝐴)‘𝑢)
43 nfcv 2904 . . . . . . . . . . . . . . . . 17 𝑧𝐵
4442, 43nfmpt 5256 . . . . . . . . . . . . . . . 16 𝑧(𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
4544nfrn 5952 . . . . . . . . . . . . . . 15 𝑧ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
4642, 45nfun 4166 . . . . . . . . . . . . . 14 𝑧((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
47 rdgeq1 8411 . . . . . . . . . . . . . . 15 (𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵))) → rec(𝐹, 𝐴) = rec((𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵))), 𝐴))
4821, 47ax-mp 5 . . . . . . . . . . . . . 14 rec(𝐹, 𝐴) = rec((𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵))), 𝐴)
49 id 22 . . . . . . . . . . . . . . 15 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → 𝑧 = (rec(𝐹, 𝐴)‘𝑢))
5032nfeq2 2921 . . . . . . . . . . . . . . . . 17 𝑦 𝑧 = (rec(𝐹, 𝐴)‘𝑢)
51 eqidd 2734 . . . . . . . . . . . . . . . . 17 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → 𝐵 = 𝐵)
5250, 49, 51mpteq12df 5235 . . . . . . . . . . . . . . . 16 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → (𝑦𝑧𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
5352rneqd 5938 . . . . . . . . . . . . . . 15 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → ran (𝑦𝑧𝐵) = ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
5449, 53uneq12d 4165 . . . . . . . . . . . . . 14 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → (𝑧 ∪ ran (𝑦𝑧𝐵)) = ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
5537, 38, 46, 48, 54rdgsucmptf 8428 . . . . . . . . . . . . 13 ((𝑢 ∈ On ∧ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)) ∈ V) → (rec(𝐹, 𝐴)‘suc 𝑢) = ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
5636, 55mpan2 690 . . . . . . . . . . . 12 (𝑢 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑢) = ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
5756eleq2d 2820 . . . . . . . . . . 11 (𝑢 ∈ On → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) ↔ 𝐵 ∈ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))))
5819, 57imbitrrid 245 . . . . . . . . . 10 (𝑢 ∈ On → ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵𝑊) → 𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢)))
5915, 58syl 17 . . . . . . . . 9 (𝑢 ∈ ω → ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵𝑊) → 𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢)))
60 rdgellim 36257 . . . . . . . . . 10 (((ω ∈ On ∧ Lim ω) ∧ suc 𝑢 ∈ ω) → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
613, 4, 60mpanl12 701 . . . . . . . . 9 (suc 𝑢 ∈ ω → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
6214, 59, 61sylsyld 61 . . . . . . . 8 (𝑢 ∈ ω → ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵𝑊) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
6362expd 417 . . . . . . 7 (𝑢 ∈ ω → (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) → (𝐵𝑊𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
6463com3r 87 . . . . . 6 (𝐵𝑊 → (𝑢 ∈ ω → (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
6564rexlimdv 3154 . . . . 5 (𝐵𝑊 → (∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
6613, 65biimtrid 241 . . . 4 (𝐵𝑊 → (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
6766alimi 1814 . . 3 (∀𝑦 𝐵𝑊 → ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
68 df-ral 3063 . . 3 (∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
6967, 68sylibr 233 . 2 (∀𝑦 𝐵𝑊 → ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω))
70 fvex 6905 . . 3 (rec(𝐹, 𝐴)‘ω) ∈ V
71 sseq2 4009 . . . 4 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐴𝑥𝐴 ⊆ (rec(𝐹, 𝐴)‘ω)))
72 nfcv 2904 . . . . . . . 8 𝑦ω
7330, 72nffv 6902 . . . . . . 7 𝑦(rec(𝐹, 𝐴)‘ω)
7473nfeq2 2921 . . . . . 6 𝑦 𝑥 = (rec(𝐹, 𝐴)‘ω)
75 eleq2 2823 . . . . . . 7 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝑦𝑥𝑦 ∈ (rec(𝐹, 𝐴)‘ω)))
76 eleq2 2823 . . . . . . 7 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐵𝑥𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
7775, 76imbi12d 345 . . . . . 6 (𝑥 = (rec(𝐹, 𝐴)‘ω) → ((𝑦𝑥𝐵𝑥) ↔ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
7874, 77albid 2216 . . . . 5 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦(𝑦𝑥𝐵𝑥) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
79 df-ral 3063 . . . . 5 (∀𝑦𝑥 𝐵𝑥 ↔ ∀𝑦(𝑦𝑥𝐵𝑥))
8078, 79, 683bitr4g 314 . . . 4 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦𝑥 𝐵𝑥 ↔ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
8171, 80anbi12d 632 . . 3 (𝑥 = (rec(𝐹, 𝐴)‘ω) → ((𝐴𝑥 ∧ ∀𝑦𝑥 𝐵𝑥) ↔ (𝐴 ⊆ (rec(𝐹, 𝐴)‘ω) ∧ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
8270, 81spcev 3597 . 2 ((𝐴 ⊆ (rec(𝐹, 𝐴)‘ω) ∧ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω)) → ∃𝑥(𝐴𝑥 ∧ ∀𝑦𝑥 𝐵𝑥))
838, 69, 82syl2an 597 1 ((𝐴𝑉 ∧ ∀𝑦 𝐵𝑊) → ∃𝑥(𝐴𝑥 ∧ ∀𝑦𝑥 𝐵𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  wral 3062  wrex 3071  Vcvv 3475  cun 3947  wss 3949  c0 4323   ciun 4998  cmpt 5232  ran crn 5678  Oncon0 6365  Lim wlim 6366  suc csuc 6367  cfv 6544  ωcom 7855  reccrdg 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410
This theorem is referenced by:  exrecfn  36261
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