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Theorem exrecfnlem 35069
 Description: Lemma for exrecfn 35070. (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 8074 . . 3 (𝐴𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
2 peano1 7601 . . . 4 ∅ ∈ ω
3 omelon 9135 . . . . 5 ω ∈ On
4 limom 7595 . . . . 5 Lim ω
5 rdglimss 35067 . . . . 5 (((ω ∈ On ∧ Lim ω) ∧ ∅ ∈ ω) → (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω))
63, 4, 5mpanl12 702 . . . 4 (∅ ∈ ω → (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω))
72, 6ax-mp 5 . . 3 (rec(𝐹, 𝐴)‘∅) ⊆ (rec(𝐹, 𝐴)‘ω)
81, 7eqsstrrdi 3948 . 2 (𝐴𝑉𝐴 ⊆ (rec(𝐹, 𝐴)‘ω))
9 rdglim2a 8080 . . . . . . . 8 ((ω ∈ On ∧ Lim ω) → (rec(𝐹, 𝐴)‘ω) = 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
103, 4, 9mp2an 692 . . . . . . 7 (rec(𝐹, 𝐴)‘ω) = 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢)
1110eleq2i 2844 . . . . . 6 (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ 𝑦 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
12 eliun 4888 . . . . . 6 (𝑦 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢) ↔ ∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢))
1311, 12bitri 278 . . . . 5 (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ ∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢))
14 peano2 7602 . . . . . . . . 9 (𝑢 ∈ ω → suc 𝑢 ∈ ω)
15 nnon 7586 . . . . . . . . . 10 (𝑢 ∈ ω → 𝑢 ∈ On)
16 eqid 2759 . . . . . . . . . . . . 13 (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
1716elrnmpt1 5800 . . . . . . . . . . . 12 ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵𝑊) → 𝐵 ∈ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
18 elun2 4083 . . . . . . . . . . . 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 6672 . . . . . . . . . . . . . 14 (rec(𝐹, 𝐴)‘𝑢) ∈ V
21 exrecfnlem.1 . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
22 nfcv 2920 . . . . . . . . . . . . . . . . . . . . 21 𝑦V
23 nfcv 2920 . . . . . . . . . . . . . . . . . . . . . 22 𝑦𝑧
24 nfmpt1 5131 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦(𝑦𝑧𝐵)
2524nfrn 5794 . . . . . . . . . . . . . . . . . . . . . 22 𝑦ran (𝑦𝑧𝐵)
2623, 25nfun 4071 . . . . . . . . . . . . . . . . . . . . 21 𝑦(𝑧 ∪ ran (𝑦𝑧𝐵))
2722, 26nfmpt 5130 . . . . . . . . . . . . . . . . . . . 20 𝑦(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
2821, 27nfcxfr 2918 . . . . . . . . . . . . . . . . . . 19 𝑦𝐹
29 nfcv 2920 . . . . . . . . . . . . . . . . . . 19 𝑦𝐴
3028, 29nfrdg 8061 . . . . . . . . . . . . . . . . . 18 𝑦rec(𝐹, 𝐴)
31 nfcv 2920 . . . . . . . . . . . . . . . . . 18 𝑦𝑢
3230, 31nffv 6669 . . . . . . . . . . . . . . . . 17 𝑦(rec(𝐹, 𝐴)‘𝑢)
3332mptexgf 6977 . . . . . . . . . . . . . . . 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 7623 . . . . . . . . . . . . . 14 ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V
3620, 35unex 7468 . . . . . . . . . . . . 13 ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)) ∈ V
37 nfcv 2920 . . . . . . . . . . . . . 14 𝑧𝐴
38 nfcv 2920 . . . . . . . . . . . . . 14 𝑧𝑢
39 nfmpt1 5131 . . . . . . . . . . . . . . . . . 18 𝑧(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
4021, 39nfcxfr 2918 . . . . . . . . . . . . . . . . 17 𝑧𝐹
4140, 37nfrdg 8061 . . . . . . . . . . . . . . . 16 𝑧rec(𝐹, 𝐴)
4241, 38nffv 6669 . . . . . . . . . . . . . . 15 𝑧(rec(𝐹, 𝐴)‘𝑢)
43 nfcv 2920 . . . . . . . . . . . . . . . . 17 𝑧𝐵
4442, 43nfmpt 5130 . . . . . . . . . . . . . . . 16 𝑧(𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
4544nfrn 5794 . . . . . . . . . . . . . . 15 𝑧ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
4642, 45nfun 4071 . . . . . . . . . . . . . 14 𝑧((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
47 rdgeq1 8058 . . . . . . . . . . . . . . 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 2937 . . . . . . . . . . . . . . . . 17 𝑦 𝑧 = (rec(𝐹, 𝐴)‘𝑢)
51 eqidd 2760 . . . . . . . . . . . . . . . . 17 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → 𝐵 = 𝐵)
5250, 49, 51mpteq12df 5115 . . . . . . . . . . . . . . . 16 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → (𝑦𝑧𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
5352rneqd 5780 . . . . . . . . . . . . . . 15 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → ran (𝑦𝑧𝐵) = ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
5449, 53uneq12d 4070 . . . . . . . . . . . . . 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 8075 . . . . . . . . . . . . 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 691 . . . . . . . . . . . 12 (𝑢 ∈ On → (rec(𝐹, 𝐴)‘suc 𝑢) = ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)))
5756eleq2d 2838 . . . . . . . . . . 11 (𝑢 ∈ On → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) ↔ 𝐵 ∈ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))))
5819, 57syl5ibr 249 . . . . . . . . . 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 35066 . . . . . . . . . 10 (((ω ∈ On ∧ Lim ω) ∧ suc 𝑢 ∈ ω) → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
613, 4, 60mpanl12 702 . . . . . . . . 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 420 . . . . . . 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 3208 . . . . 5 (𝐵𝑊 → (∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
6613, 65syl5bi 245 . . . 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 3076 . . 3 (∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
6967, 68sylibr 237 . 2 (∀𝑦 𝐵𝑊 → ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω))
70 fvex 6672 . . 3 (rec(𝐹, 𝐴)‘ω) ∈ V
71 sseq2 3919 . . . 4 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐴𝑥𝐴 ⊆ (rec(𝐹, 𝐴)‘ω)))
72 nfcv 2920 . . . . . . . 8 𝑦ω
7330, 72nffv 6669 . . . . . . 7 𝑦(rec(𝐹, 𝐴)‘ω)
7473nfeq2 2937 . . . . . 6 𝑦 𝑥 = (rec(𝐹, 𝐴)‘ω)
75 eleq2 2841 . . . . . . 7 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝑦𝑥𝑦 ∈ (rec(𝐹, 𝐴)‘ω)))
76 eleq2 2841 . . . . . . 7 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐵𝑥𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
7775, 76imbi12d 349 . . . . . 6 (𝑥 = (rec(𝐹, 𝐴)‘ω) → ((𝑦𝑥𝐵𝑥) ↔ (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
7874, 77albid 2223 . . . . 5 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦(𝑦𝑥𝐵𝑥) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
79 df-ral 3076 . . . . 5 (∀𝑦𝑥 𝐵𝑥 ↔ ∀𝑦(𝑦𝑥𝐵𝑥))
8078, 79, 683bitr4g 318 . . . 4 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦𝑥 𝐵𝑥 ↔ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
8171, 80anbi12d 634 . . 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 3526 . 2 ((𝐴 ⊆ (rec(𝐹, 𝐴)‘ω) ∧ ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω)) → ∃𝑥(𝐴𝑥 ∧ ∀𝑦𝑥 𝐵𝑥))
838, 69, 82syl2an 599 1 ((𝐴𝑉 ∧ ∀𝑦 𝐵𝑊) → ∃𝑥(𝐴𝑥 ∧ ∀𝑦𝑥 𝐵𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2112  ∀wral 3071  ∃wrex 3072  Vcvv 3410   ∪ cun 3857   ⊆ wss 3859  ∅c0 4226  ∪ ciun 4884   ↦ cmpt 5113  ran crn 5526  Oncon0 6170  Lim wlim 6171  suc csuc 6172  ‘cfv 6336  ωcom 7580  reccrdg 8056 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460  ax-inf2 9130 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7581  df-wrecs 7958  df-recs 8019  df-rdg 8057 This theorem is referenced by:  exrecfn  35070
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