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Theorem tfrlemiex 5945
 Description: Lemma for tfrlemi1 5946. (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 5943 . . 3 (𝜑𝐵 ∈ V)
7 uniexg 4175 . . 3 (𝐵 ∈ V → 𝐵 ∈ V)
86, 7syl 14 . 2 (𝜑 𝐵 ∈ V)
91, 2, 3, 4, 5tfrlemibfn 5942 . . 3 (𝜑 𝐵 Fn 𝑥)
101, 2, 3, 4, 5tfrlemiubacc 5944 . . 3 (𝜑 → ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))
119, 10jca 290 . 2 (𝜑 → ( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
12 fneq1 4987 . . . 4 (𝑓 = 𝐵 → (𝑓 Fn 𝑥 𝐵 Fn 𝑥))
13 fveq1 5177 . . . . . 6 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
14 reseq1 4606 . . . . . . 7 (𝑓 = 𝐵 → (𝑓𝑢) = ( 𝐵𝑢))
1514fveq2d 5182 . . . . . 6 (𝑓 = 𝐵 → (𝐹‘(𝑓𝑢)) = (𝐹‘( 𝐵𝑢)))
1613, 15eqeq12d 2054 . . . . 5 (𝑓 = 𝐵 → ((𝑓𝑢) = (𝐹‘(𝑓𝑢)) ↔ ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
1716ralbidv 2326 . . . 4 (𝑓 = 𝐵 → (∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢)) ↔ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))))
1812, 17anbi12d 442 . . 3 (𝑓 = 𝐵 → ((𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))) ↔ ( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢)))))
1918spcegv 2641 . 2 ( 𝐵 ∈ V → (( 𝐵 Fn 𝑥 ∧ ∀𝑢𝑥 ( 𝐵𝑢) = (𝐹‘( 𝐵𝑢))) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢)))))
208, 11, 19sylc 56 1 (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢𝑥 (𝑓𝑢) = (𝐹‘(𝑓𝑢))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 885  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  Vcvv 2557   ∪ cun 2915  {csn 3375  ⟨cop 3378  ∪ cuni 3580  Oncon0 4100   ↾ cres 4347  Fun wfun 4896   Fn wfn 4897  ‘cfv 4902 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920 This theorem is referenced by:  tfrlemi1  5946
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