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Theorem tfrlem3a 7433
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 5952 . . . 4 ((𝑓 = 𝐺𝑥 = 𝑧) → (𝑓 Fn 𝑥𝐺 Fn 𝑧))
3 simpll 789 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐺)
4 simpr 477 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
53, 4fveq12d 6164 . . . . . 6 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓𝑦) = (𝐺𝑤))
63, 4reseq12d 5367 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓𝑦) = (𝐺𝑤))
76fveq2d 6162 . . . . . 6 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝐺𝑤)))
85, 7eqeq12d 2636 . . . . 5 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
9 simplr 791 . . . . 5 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
108, 9cbvraldva2 3167 . . . 4 ((𝑓 = 𝐺𝑥 = 𝑧) → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
112, 10anbi12d 746 . . 3 ((𝑓 = 𝐺𝑥 = 𝑧) → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
1211cbvrexdva 3170 . 2 (𝑓 = 𝐺 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
13 tfrlem3.1 . 2 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
141, 12, 13elab2 3342 1 (𝐺𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2908  wrex 2909  Vcvv 3190  cres 5086  Oncon0 5692   Fn wfn 5852  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-res 5096  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865
This theorem is referenced by:  tfrlem3  7434  tfrlem5  7436  tfrlem9a  7442
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