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Theorem tfrlem3ag 6206
 Description: Lemma for transfinite recursion. This lemma just changes some bound variables in 𝐴 for later use. (Contributed by Jim Kingdon, 5-Jul-2019.)
Hypothesis
Ref Expression
tfrlem3.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
Assertion
Ref Expression
tfrlem3ag (𝐺 ∈ V → (𝐺𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
Distinct variable groups:   𝑤,𝑓,𝑥,𝑦,𝑧,𝐹   𝑓,𝐺,𝑤,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑤,𝑓)

Proof of Theorem tfrlem3ag
StepHypRef Expression
1 fneq12 5216 . . . 4 ((𝑓 = 𝐺𝑥 = 𝑧) → (𝑓 Fn 𝑥𝐺 Fn 𝑧))
2 simpll 518 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐺)
3 simpr 109 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
42, 3fveq12d 5428 . . . . . 6 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓𝑦) = (𝐺𝑤))
52, 3reseq12d 4820 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓𝑦) = (𝐺𝑤))
65fveq2d 5425 . . . . . 6 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝐺𝑤)))
74, 6eqeq12d 2154 . . . . 5 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
8 simplr 519 . . . . 5 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
97, 8cbvraldva2 2661 . . . 4 ((𝑓 = 𝐺𝑥 = 𝑧) → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
101, 9anbi12d 464 . . 3 ((𝑓 = 𝐺𝑥 = 𝑧) → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
1110cbvrexdva 2664 . 2 (𝑓 = 𝐺 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
12 tfrlem3.1 . 2 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
1311, 12elab2g 2831 1 (𝐺 ∈ V → (𝐺𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417  Vcvv 2686  Oncon0 4285   ↾ cres 4541   Fn wfn 5118  ‘cfv 5123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131 This theorem is referenced by:  tfrlemisucaccv  6222
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