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Theorem exrecfnlem 35850
Description: Lemma for exrecfn 35851. (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 8373 . . 3 (𝐴𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
2 peano1 7825 . . . 4 ∅ ∈ ω
3 omelon 9582 . . . . 5 ω ∈ On
4 limom 7818 . . . . 5 Lim ω
5 rdglimss 35848 . . . . 5 (((ω ∈ On ∧ Lim ω) ∧ ∅ ∈ ω) → (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω))
63, 4, 5mpanl12 700 . . . 4 (∅ ∈ ω → (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω))
72, 6ax-mp 5 . . 3 (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω)
81, 7eqsstrrdi 3999 . 2 (𝐴𝑉𝐴 ⊆ (rec(𝐹, 𝐴)‘ω))
9 rdglim2a 8379 . . . . . . . 8 ((ω ∈ On ∧ Lim ω) → (rec(𝐹, 𝐴)‘ω) = 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
103, 4, 9mp2an 690 . . . . . . 7 (rec(𝐹, 𝐴)‘ω) = 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢)
1110eleq2i 2829 . . . . . 6 (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ 𝑦 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
12 eliun 4958 . . . . . 6 (𝑦 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢) ↔ ∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢))
1311, 12bitri 274 . . . . 5 (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ ∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢))
14 peano2 7827 . . . . . . . . 9 (𝑢 ∈ ω → suc 𝑢 ∈ ω)
15 nnon 7808 . . . . . . . . . 10 (𝑢 ∈ ω → 𝑢 ∈ On)
16 eqid 2736 . . . . . . . . . . . . 13 (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
1716elrnmpt1 5913 . . . . . . . . . . . 12 ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵𝑊) → 𝐵 ∈ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
18 elun2 4137 . . . . . . . . . . . 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 6855 . . . . . . . . . . . . . 14 (rec(𝐹, 𝐴)‘𝑢) ∈ V
21 exrecfnlem.1 . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
22 nfcv 2907 . . . . . . . . . . . . . . . . . . . . 21 𝑦V
23 nfcv 2907 . . . . . . . . . . . . . . . . . . . . . 22 𝑦𝑧
24 nfmpt1 5213 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦(𝑦𝑧𝐵)
2524nfrn 5907 . . . . . . . . . . . . . . . . . . . . . 22 𝑦ran (𝑦𝑧𝐵)
2623, 25nfun 4125 . . . . . . . . . . . . . . . . . . . . 21 𝑦(𝑧 ∪ ran (𝑦𝑧𝐵))
2722, 26nfmpt 5212 . . . . . . . . . . . . . . . . . . . 20 𝑦(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
2821, 27nfcxfr 2905 . . . . . . . . . . . . . . . . . . 19 𝑦𝐹
29 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑦𝐴
3028, 29nfrdg 8360 . . . . . . . . . . . . . . . . . 18 𝑦rec(𝐹, 𝐴)
31 nfcv 2907 . . . . . . . . . . . . . . . . . 18 𝑦𝑢
3230, 31nffv 6852 . . . . . . . . . . . . . . . . 17 𝑦(rec(𝐹, 𝐴)‘𝑢)
3332mptexgf 7172 . . . . . . . . . . . . . . . 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 7849 . . . . . . . . . . . . . 14 ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V
3620, 35unex 7680 . . . . . . . . . . . . 13 ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)) ∈ V
37 nfcv 2907 . . . . . . . . . . . . . 14 𝑧𝐴
38 nfcv 2907 . . . . . . . . . . . . . 14 𝑧𝑢
39 nfmpt1 5213 . . . . . . . . . . . . . . . . . 18 𝑧(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
4021, 39nfcxfr 2905 . . . . . . . . . . . . . . . . 17 𝑧𝐹
4140, 37nfrdg 8360 . . . . . . . . . . . . . . . 16 𝑧rec(𝐹, 𝐴)
4241, 38nffv 6852 . . . . . . . . . . . . . . 15 𝑧(rec(𝐹, 𝐴)‘𝑢)
43 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑧𝐵
4442, 43nfmpt 5212 . . . . . . . . . . . . . . . 16 𝑧(𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
4544nfrn 5907 . . . . . . . . . . . . . . 15 𝑧ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
4642, 45nfun 4125 . . . . . . . . . . . . . 14 𝑧((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
47 rdgeq1 8357 . . . . . . . . . . . . . . 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 2924 . . . . . . . . . . . . . . . . 17 𝑦 𝑧 = (rec(𝐹, 𝐴)‘𝑢)
51 eqidd 2737 . . . . . . . . . . . . . . . . 17 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → 𝐵 = 𝐵)
5250, 49, 51mpteq12df 5191 . . . . . . . . . . . . . . . 16 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → (𝑦𝑧𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
5352rneqd 5893 . . . . . . . . . . . . . . 15 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → ran (𝑦𝑧𝐵) = ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
5449, 53uneq12d 4124 . . . . . . . . . . . . . 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 8374 . . . . . . . . . . . . 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 689 . . . . . . . . . . . 12 (𝑢 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑢) = ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
5756eleq2d 2823 . . . . . . . . . . 11 (𝑢 ∈ On → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) ↔ 𝐵 ∈ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))))
5819, 57syl5ibr 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 35847 . . . . . . . . . 10 (((ω ∈ On ∧ Lim ω) ∧ suc 𝑢 ∈ ω) → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
613, 4, 60mpanl12 700 . . . . . . . . 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 416 . . . . . . 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 3150 . . . . 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 1813 . . 3 (∀𝑦 𝐵𝑊 → ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
68 df-ral 3065 . . 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 6855 . . 3 (rec(𝐹, 𝐴)‘ω) ∈ V
71 sseq2 3970 . . . 4 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐴𝑥𝐴 ⊆ (rec(𝐹, 𝐴)‘ω)))
72 nfcv 2907 . . . . . . . 8 𝑦ω
7330, 72nffv 6852 . . . . . . 7 𝑦(rec(𝐹, 𝐴)‘ω)
7473nfeq2 2924 . . . . . 6 𝑦 𝑥 = (rec(𝐹, 𝐴)‘ω)
75 eleq2 2826 . . . . . . 7 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝑦𝑥𝑦 ∈ (rec(𝐹, 𝐴)‘ω)))
76 eleq2 2826 . . . . . . 7 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐵𝑥𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
7775, 76imbi12d 344 . . . . . 6 (𝑥 = (rec(𝐹, 𝐴)‘ω) → ((𝑦𝑥𝐵𝑥) ↔ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
7874, 77albid 2215 . . . . 5 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦(𝑦𝑥𝐵𝑥) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
79 df-ral 3065 . . . . 5 (∀𝑦𝑥 𝐵𝑥 ↔ ∀𝑦(𝑦𝑥𝐵𝑥))
8078, 79, 683bitr4g 313 . . . 4 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦𝑥 𝐵𝑥 ↔ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
8171, 80anbi12d 631 . . 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 3565 . 2 ((𝐴 ⊆ (rec(𝐹, 𝐴)‘ω) ∧ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω)) → ∃𝑥(𝐴𝑥 ∧ ∀𝑦𝑥 𝐵𝑥))
838, 69, 82syl2an 596 1 ((𝐴𝑉 ∧ ∀𝑦 𝐵𝑊) → ∃𝑥(𝐴𝑥 ∧ ∀𝑦𝑥 𝐵𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  cun 3908  wss 3910  c0 4282   ciun 4954  cmpt 5188  ran crn 5634  Oncon0 6317  Lim wlim 6318  suc csuc 6319  cfv 6496  ωcom 7802  reccrdg 8355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356
This theorem is referenced by:  exrecfn  35851
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