Theorem tfrlem3a 7817

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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlem3.2	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
tfrlem3a	⊢ (𝐺 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))

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

Step	Hyp	Ref	Expression
1		tfrlem3.2	. 2 ⊢ 𝐺 ∈ V
2		fneq12 6282	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → (𝑓 Fn 𝑥 ↔ 𝐺 Fn 𝑧))
3		simpll 754	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐺)
4		simpr 477	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
5	3, 4	fveq12d 6506	. . . . . 6 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓‘𝑦) = (𝐺‘𝑤))
6	3, 4	reseq12d 5696	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓 ↾ 𝑦) = (𝐺 ↾ 𝑤))
7	6	fveq2d 6503	. . . . . 6 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝐺 ↾ 𝑤)))
8	5, 7	eqeq12d 2794	. . . . 5 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))
9		simplr 756	. . . . 5 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
10	8, 9	cbvraldva2 3388	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))
11	2, 10	anbi12d 621	. . 3 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) ↔ (𝐺 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤)))))
12	11	cbvrexdva 3392	. 2 ⊢ (𝑓 = 𝐺 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦))) ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤)))))
13		tfrlem3.1	. 2 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
14	1, 12, 13	elab2 3586	1 ⊢ (𝐺 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))

Syntax hints:   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {cab 2759  ∀wral 3089  ∃wrex 3090  Vcvv 3416   ↾ cres 5409  Oncon0 6029   Fn wfn 6183  ‘cfv 6188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-res 5419  df-iota 6152  df-fun 6190  df-fn 6191  df-fv 6196

This theorem is referenced by:  tfrlem3  7818  tfrlem5  7820  tfrlem9a  7826