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Theorem tfrlemiex 6325
Description: Lemma for tfrlemi1 6326. (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 6323 . . 3 (𝜑𝐵 ∈ V)
7 uniexg 4435 . . 3 (𝐵 ∈ V → 𝐵 ∈ V)
86, 7syl 14 . 2 (𝜑 𝐵 ∈ V)
91, 2, 3, 4, 5tfrlemibfn 6322 . . 3 (𝜑 𝐵 Fn 𝑥)
101, 2, 3, 4, 5tfrlemiubacc 6324 . . 3 (𝜑 → ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))
119, 10jca 306 . 2 (𝜑 → ( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
12 fneq1 5299 . . . 4 (𝑓 = 𝐵 → (𝑓 Fn 𝑥 𝐵 Fn 𝑥))
13 fveq1 5509 . . . . . 6 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
14 reseq1 4896 . . . . . . 7 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
1514fveq2d 5514 . . . . . 6 (𝑓 = 𝐵 → (𝐹‘(𝑓𝑢)) = (𝐹‘( 𝐵𝑢)))
1613, 15eqeq12d 2192 . . . . 5 (𝑓 = 𝐵 → ((𝑓𝑢) = (𝐹‘(𝑓𝑢)) ↔ ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
1716ralbidv 2477 . . . 4 (𝑓 = 𝐵 → (∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢)) ↔ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
1812, 17anbi12d 473 . . 3 (𝑓 = 𝐵 → ((𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))) ↔ ( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))))
1918spcegv 2825 . 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 104  w3a 978  wal 1351   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wral 2455  wrex 2456  Vcvv 2737  cun 3127  {csn 3591  cop 3594   cuni 3807  Oncon0 4359  cres 4624  Fun wfun 5205   Fn wfn 5206  cfv 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-recs 6299
This theorem is referenced by:  tfrlemi1  6326
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