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Theorem tfrlem3a 8000
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 6423 . . . 4 ((𝑓 = 𝐺𝑥 = 𝑧) → (𝑓 Fn 𝑥𝐺 Fn 𝑧))
3 simpll 766 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐺)
4 simpr 488 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
53, 4fveq12d 6656 . . . . . 6 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓𝑦) = (𝐺𝑤))
63, 4reseq12d 5823 . . . . . . 7 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓𝑦) = (𝐺𝑤))
76fveq2d 6653 . . . . . 6 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐹‘(𝑓𝑦)) = (𝐹‘(𝐺𝑤)))
85, 7eqeq12d 2817 . . . . 5 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
9 simplr 768 . . . . 5 (((𝑓 = 𝐺𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
108, 9cbvraldva2 3406 . . . 4 ((𝑓 = 𝐺𝑥 = 𝑧) → (∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)) ↔ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
112, 10anbi12d 633 . . 3 ((𝑓 = 𝐺𝑥 = 𝑧) → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦))) ↔ (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤)))))
1211cbvrexdva 3410 . 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 3621 1 (𝐺𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤𝑧 (𝐺𝑤) = (𝐹‘(𝐺𝑤))))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  Vcvv 3444  cres 5525  Oncon0 6163   Fn wfn 6323  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-un 3889  df-in 3891  df-ss 3901  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336
This theorem is referenced by:  tfrlem3  8001  tfrlem5  8003  tfrlem9a  8009
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