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Theorem tfrlem3a 6260
Description: Lemma for transfinite recursion. Let 𝐴 be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in 𝐴 for later use. (Contributed by NM, 9-Apr-1995.)
Hypotheses
Ref Expression
tfrlem3.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlem3.2 𝐺 ∈ V
Assertion
Ref Expression
tfrlem3a (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 tfrlem3a
StepHypRef Expression
1 tfrlem3.2 . 2 𝐺 ∈ V
2 fneq12 5266 . . . 4 ((𝑓 = 𝐺𝑥 = 𝑧) → (𝑓 Fn 𝑥𝐺 Fn 𝑧))
3 simpll 519 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐺)
4 simpr 109 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
53, 4fveq12d 5478 . . . . . 6 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓𝑦) = (𝐺𝑤))
63, 4reseq12d 4870 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓𝑦) = (𝐺𝑤))
76fveq2d 5475 . . . . . 6 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝐺𝑤)))
85, 7eqeq12d 2172 . . . . 5 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
9 simpr 109 . . . . . 6 ((𝑓 = 𝐺𝑥 = 𝑧) → 𝑥 = 𝑧)
109adantr 274 . . . . 5 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
118, 10cbvraldva2 2687 . . . 4 ((𝑓 = 𝐺𝑥 = 𝑧) → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
122, 11anbi12d 465 . . 3 ((𝑓 = 𝐺𝑥 = 𝑧) → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
1312cbvrexdva 2690 . 2 (𝑓 = 𝐺 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
14 tfrlem3.1 . 2 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
151, 13, 14elab2 2860 1 (𝐺𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1335  wcel 2128  {cab 2143  wral 2435  wrex 2436  Vcvv 2712  Oncon0 4326  cres 4591   Fn wfn 5168  cfv 5173
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4029  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-res 4601  df-iota 5138  df-fun 5175  df-fn 5176  df-fv 5181
This theorem is referenced by:  tfrlem3  6261  tfrlem5  6264  tfrlemisucaccv  6275  tfrlemibxssdm  6277  tfrlemi14d  6283  tfrexlem  6284
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