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Theorem tfrlemibacc 6175
Description: Each element of 𝐵 is an acceptable function. Lemma for tfrlemi1 6181. (Contributed by Jim Kingdon, 14-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
tfrlemibacc (𝜑𝐵𝐴)
Distinct variable groups:   𝑓,𝑔,,𝑤,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,,𝑤,𝑥,𝑦,𝑧   𝜑,𝑤,𝑦   𝑤,𝐵,𝑓,𝑔,,𝑧   𝜑,𝑔,,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑓)   𝐵(𝑥,𝑦)

Proof of Theorem tfrlemibacc
StepHypRef Expression
1 tfrlemi1.3 . 2 𝐵 = { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))}
2 simpr3 970 . . . . . . 7 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))
3 tfrlemisucfn.1 . . . . . . . 8 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
4 tfrlemisucfn.2 . . . . . . . . 9 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
54ad2antrr 477 . . . . . . . 8 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
6 tfrlemi1.4 . . . . . . . . . 10 (𝜑𝑥 ∈ On)
76ad2antrr 477 . . . . . . . . 9 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑥 ∈ On)
8 simplr 502 . . . . . . . . 9 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧𝑥)
9 onelon 4264 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑧𝑥) → 𝑧 ∈ On)
107, 8, 9syl2anc 406 . . . . . . . 8 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑧 ∈ On)
11 simpr1 968 . . . . . . . 8 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔 Fn 𝑧)
12 simpr2 969 . . . . . . . 8 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝑔𝐴)
133, 5, 10, 11, 12tfrlemisucaccv 6174 . . . . . . 7 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴)
142, 13eqeltrd 2189 . . . . . 6 (((𝜑𝑧𝑥) ∧ (𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))) → 𝐴)
1514ex 114 . . . . 5 ((𝜑𝑧𝑥) → ((𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → 𝐴))
1615exlimdv 1771 . . . 4 ((𝜑𝑧𝑥) → (∃𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → 𝐴))
1716rexlimdva 2521 . . 3 (𝜑 → (∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})) → 𝐴))
1817abssdv 3135 . 2 (𝜑 → { ∣ ∃𝑧𝑥𝑔(𝑔 Fn 𝑧𝑔𝐴 = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}))} ⊆ 𝐴)
191, 18syl5eqss 3107 1 (𝜑𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 943  wal 1310   = wceq 1312  wex 1449  wcel 1461  {cab 2099  wral 2388  wrex 2389  Vcvv 2655  cun 3033  wss 3035  {csn 3491  cop 3494  Oncon0 4243  cres 4499  Fun wfun 5073   Fn wfn 5074  cfv 5079
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 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-res 4509  df-iota 5044  df-fun 5081  df-fn 5082  df-fv 5087
This theorem is referenced by:  tfrlemibfn  6177  tfrlemiubacc  6179
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