Theorem tfrlem3a 6581

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 5474	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → (𝑓 Fn 𝑥 ↔ 𝐺 Fn 𝑧))
3		simpll 531	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐺)
4		simpr 110	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
5	3, 4	fveq12d 5702	. . . . . 6 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓‘𝑦) = (𝐺‘𝑤))
6	3, 4	reseq12d 5064	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓 ↾ 𝑦) = (𝐺 ↾ 𝑤))
7	6	fveq2d 5699	. . . . . 6 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝐺 ↾ 𝑤)))
8	5, 7	eqeq12d 2253	. . . . 5 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))
9		simpr 110	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
10	9	adantr 276	. . . . 5 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
11	8, 10	cbvraldva2 2793	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))
12	2, 11	anbi12d 477	. . 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)))))
13	12	cbvrexdva 2796	. 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)))))
14		tfrlem3.1	. 2 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
15	1, 13, 14	elab2 2974	1 ⊢ (𝐺 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  Vcvv 2821  Oncon0 4508   ↾ cres 4776   Fn wfn 5372  ‘cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by:  tfrlem3  6582  tfrlem5  6585  tfrlemisucaccv  6596  tfrlemibxssdm  6598  tfrlemi14d  6604  tfrexlem  6605