Theorem tfrlem3a 8015

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 6451	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → (𝑓 Fn 𝑥 ↔ 𝐺 Fn 𝑧))
3		simpll 765	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐺)
4		simpr 487	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
5	3, 4	fveq12d 6679	. . . . . 6 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓‘𝑦) = (𝐺‘𝑤))
6	3, 4	reseq12d 5856	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓 ↾ 𝑦) = (𝐺 ↾ 𝑤))
7	6	fveq2d 6676	. . . . . 6 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝐺 ↾ 𝑤)))
8	5, 7	eqeq12d 2839	. . . . 5 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))
9		simplr 767	. . . . 5 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
10	8, 9	cbvraldva2 3458	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))
11	2, 10	anbi12d 632	. . 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 3462	. 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 3672	1 ⊢ (𝐺 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2801  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ↾ cres 5559  Oncon0 6193   Fn wfn 6352  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-res 5569  df-iota 6316  df-fun 6359  df-fn 6360  df-fv 6365

This theorem is referenced by:  tfrlem3  8016  tfrlem5  8018  tfrlem9a  8024