Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  exrecfnlem Structured version   Visualization version   GIF version

Theorem exrecfnlem 37380
Description: Lemma for exrecfn 37381. (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 8467 . . 3 (𝐴𝑉 → (rec(𝐹, 𝐴)‘∅) = 𝐴)
2 peano1 7910 . . . 4 ∅ ∈ ω
3 omelon 9686 . . . . 5 ω ∈ On
4 limom 7903 . . . . 5 Lim ω
5 rdglimss 37378 . . . . 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 4029 . 2 (𝐴𝑉𝐴 ⊆ (rec(𝐹, 𝐴)‘ω))
9 rdglim2a 8473 . . . . . . . 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 2833 . . . . . 6 (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) ↔ 𝑦 𝑢 ∈ ω (rec(𝐹, 𝐴)‘𝑢))
12 eliun 4995 . . . . . 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 7912 . . . . . . . . 9 (𝑢 ∈ ω → suc 𝑢 ∈ ω)
15 nnon 7893 . . . . . . . . . 10 (𝑢 ∈ ω → 𝑢 ∈ On)
16 eqid 2737 . . . . . . . . . . . . 13 (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
1716elrnmpt1 5971 . . . . . . . . . . . 12 ((𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ∧ 𝐵𝑊) → 𝐵 ∈ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
18 elun2 4183 . . . . . . . . . . . 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 6919 . . . . . . . . . . . . . 14 (rec(𝐹, 𝐴)‘𝑢) ∈ V
21 exrecfnlem.1 . . . . . . . . . . . . . . . . . . . 20 𝐹 = (𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
22 nfcv 2905 . . . . . . . . . . . . . . . . . . . . 21 𝑦V
23 nfcv 2905 . . . . . . . . . . . . . . . . . . . . . 22 𝑦𝑧
24 nfmpt1 5250 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦(𝑦𝑧𝐵)
2524nfrn 5963 . . . . . . . . . . . . . . . . . . . . . 22 𝑦ran (𝑦𝑧𝐵)
2623, 25nfun 4170 . . . . . . . . . . . . . . . . . . . . 21 𝑦(𝑧 ∪ ran (𝑦𝑧𝐵))
2722, 26nfmpt 5249 . . . . . . . . . . . . . . . . . . . 20 𝑦(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
2821, 27nfcxfr 2903 . . . . . . . . . . . . . . . . . . 19 𝑦𝐹
29 nfcv 2905 . . . . . . . . . . . . . . . . . . 19 𝑦𝐴
3028, 29nfrdg 8454 . . . . . . . . . . . . . . . . . 18 𝑦rec(𝐹, 𝐴)
31 nfcv 2905 . . . . . . . . . . . . . . . . . 18 𝑦𝑢
3230, 31nffv 6916 . . . . . . . . . . . . . . . . 17 𝑦(rec(𝐹, 𝐴)‘𝑢)
3332mptexgf 7242 . . . . . . . . . . . . . . . 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 7932 . . . . . . . . . . . . . 14 ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵) ∈ V
3620, 35unex 7764 . . . . . . . . . . . . 13 ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)) ∈ V
37 nfcv 2905 . . . . . . . . . . . . . 14 𝑧𝐴
38 nfcv 2905 . . . . . . . . . . . . . 14 𝑧𝑢
39 nfmpt1 5250 . . . . . . . . . . . . . . . . . 18 𝑧(𝑧 ∈ V ↦ (𝑧 ∪ ran (𝑦𝑧𝐵)))
4021, 39nfcxfr 2903 . . . . . . . . . . . . . . . . 17 𝑧𝐹
4140, 37nfrdg 8454 . . . . . . . . . . . . . . . 16 𝑧rec(𝐹, 𝐴)
4241, 38nffv 6916 . . . . . . . . . . . . . . 15 𝑧(rec(𝐹, 𝐴)‘𝑢)
43 nfcv 2905 . . . . . . . . . . . . . . . . 17 𝑧𝐵
4442, 43nfmpt 5249 . . . . . . . . . . . . . . . 16 𝑧(𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
4544nfrn 5963 . . . . . . . . . . . . . . 15 𝑧ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵)
4642, 45nfun 4170 . . . . . . . . . . . . . 14 𝑧((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
47 rdgeq1 8451 . . . . . . . . . . . . . . 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 2923 . . . . . . . . . . . . . . . . 17 𝑦 𝑧 = (rec(𝐹, 𝐴)‘𝑢)
51 eqidd 2738 . . . . . . . . . . . . . . . . 17 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → 𝐵 = 𝐵)
5250, 49, 51mpteq12df 5228 . . . . . . . . . . . . . . . 16 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → (𝑦𝑧𝐵) = (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
5352rneqd 5949 . . . . . . . . . . . . . . 15 (𝑧 = (rec(𝐹, 𝐴)‘𝑢) → ran (𝑦𝑧𝐵) = ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))
5449, 53uneq12d 4169 . . . . . . . . . . . . . 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 8468 . . . . . . . . . . . . 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 2827 . . . . . . . . . . 11 (𝑢 ∈ On → (𝐵 ∈ (rec(𝐹, 𝐴)‘suc 𝑢) ↔ 𝐵 ∈ ((rec(𝐹, 𝐴)‘𝑢) ∪ ran (𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) ↦ 𝐵))))
5819, 57imbitrrid 246 . . . . . . . . . 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 37377 . . . . . . . . . 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 415 . . . . . . 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 3153 . . . . 5 (𝐵𝑊 → (∃𝑢 ∈ ω 𝑦 ∈ (rec(𝐹, 𝐴)‘𝑢) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
6613, 65biimtrid 242 . . . 4 (𝐵𝑊 → (𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
6766alimi 1811 . . 3 (∀𝑦 𝐵𝑊 → ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
68 df-ral 3062 . . 3 (∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω)))
6967, 68sylibr 234 . 2 (∀𝑦 𝐵𝑊 → ∀𝑦 ∈ (rec(𝐹, 𝐴)‘ω)𝐵 ∈ (rec(𝐹, 𝐴)‘ω))
70 fvex 6919 . . 3 (rec(𝐹, 𝐴)‘ω) ∈ V
71 sseq2 4010 . . . 4 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝐴𝑥𝐴 ⊆ (rec(𝐹, 𝐴)‘ω)))
72 nfcv 2905 . . . . . . . 8 𝑦ω
7330, 72nffv 6916 . . . . . . 7 𝑦(rec(𝐹, 𝐴)‘ω)
7473nfeq2 2923 . . . . . 6 𝑦 𝑥 = (rec(𝐹, 𝐴)‘ω)
75 eleq2 2830 . . . . . . 7 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (𝑦𝑥𝑦 ∈ (rec(𝐹, 𝐴)‘ω)))
76 eleq2 2830 . . . . . . 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 2222 . . . . 5 (𝑥 = (rec(𝐹, 𝐴)‘ω) → (∀𝑦(𝑦𝑥𝐵𝑥) ↔ ∀𝑦(𝑦 ∈ (rec(𝐹, 𝐴)‘ω) → 𝐵 ∈ (rec(𝐹, 𝐴)‘ω))))
79 df-ral 3062 . . . . 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 3606 . 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 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cun 3949  wss 3951  c0 4333   ciun 4991  cmpt 5225  ran crn 5686  Oncon0 6384  Lim wlim 6385  suc csuc 6386  cfv 6561  ωcom 7887  reccrdg 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  exrecfn  37381
  Copyright terms: Public domain W3C validator