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Theorem tfrlemiex 6110
Description: Lemma for tfrlemi1 6111. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlemisucfn.2 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
tfrlemi1.3 𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}
tfrlemi1.4 (𝜑𝑥 ∈ On)
tfrlemi1.5 (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
Assertion
Ref Expression
tfrlemiex (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))))
Distinct variable groups:   𝑓,𝑔,,𝑢,𝑤,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,,𝑢,𝑤,𝑥,𝑦,𝑧   𝜑,𝑤,𝑦   𝑢,𝐵,𝑤,𝑓,𝑔,,𝑧   𝜑,𝑔,,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑢,𝑓)   𝐵(𝑥,𝑦)

Proof of Theorem tfrlemiex
StepHypRef Expression
1 tfrlemisucfn.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
2 tfrlemisucfn.2 . . . 4 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
3 tfrlemi1.3 . . . 4 𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}
4 tfrlemi1.4 . . . 4 (𝜑𝑥 ∈ On)
5 tfrlemi1.5 . . . 4 (𝜑 → ∀𝑧𝑥𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤𝑧 (𝑔𝑤) = (𝐹‘(𝑔𝑤))))
61, 2, 3, 4, 5tfrlemibex 6108 . . 3 (𝜑𝐵 ∈ V)
7 uniexg 4275 . . 3 (𝐵 ∈ V → 𝐵 ∈ V)
86, 7syl 14 . 2 (𝜑 𝐵 ∈ V)
91, 2, 3, 4, 5tfrlemibfn 6107 . . 3 (𝜑 𝐵 Fn 𝑥)
101, 2, 3, 4, 5tfrlemiubacc 6109 . . 3 (𝜑 → ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))
119, 10jca 301 . 2 (𝜑 → ( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
12 fneq1 5115 . . . 4 (𝑓 = 𝐵 → (𝑓 Fn 𝑥 𝐵 Fn 𝑥))
13 fveq1 5317 . . . . . 6 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
14 reseq1 4720 . . . . . . 7 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
1514fveq2d 5322 . . . . . 6 (𝑓 = 𝐵 → (𝐹‘(𝑓𝑢)) = (𝐹‘( 𝐵𝑢)))
1613, 15eqeq12d 2103 . . . . 5 (𝑓 = 𝐵 → ((𝑓𝑢) = (𝐹‘(𝑓𝑢)) ↔ ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
1716ralbidv 2381 . . . 4 (𝑓 = 𝐵 → (∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢)) ↔ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
1812, 17anbi12d 458 . . 3 (𝑓 = 𝐵 → ((𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))) ↔ ( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))))
1918spcegv 2708 . 2 ( 𝐵 ∈ V → (( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢)))))
208, 11, 19sylc 62 1 (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 925  wal 1288   = wceq 1290  wex 1427  wcel 1439  {cab 2075  wral 2360  wrex 2361  Vcvv 2620  cun 2998  {csn 3450  cop 3453   cuni 3659  Oncon0 4199  cres 4453  Fun wfun 5022   Fn wfn 5023  cfv 5028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-recs 6084
This theorem is referenced by:  tfrlemi1  6111
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