Theorem tfr1onlem3ag 6234

Description: Lemma for transfinite recursion. This lemma changes some bound variables in 𝐴 (version of tfrlem3ag 6206 but for tfr1on 6247 related lemmas). (Contributed by Jim Kingdon, 13-Mar-2022.)

Hypothesis

Ref	Expression
tfr1onlem3ag.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}

Assertion

Ref	Expression
tfr1onlem3ag	⊢ (𝐻 ∈ 𝑉 → (𝐻 ∈ 𝐴 ↔ ∃𝑧 ∈ 𝑋 (𝐻 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤)))))

Distinct variable groups:   𝑓,𝐺,𝑤,𝑥,𝑦,𝑧   𝑓,𝐻,𝑤,𝑥,𝑦,𝑧   𝑓,𝑋,𝑥,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧,𝑤,𝑓)   𝑉(𝑥,𝑦,𝑧,𝑤,𝑓)   𝑋(𝑦,𝑤)

Proof of Theorem tfr1onlem3ag

Step	Hyp	Ref	Expression
1		fneq12 5216	. . . 4 ⊢ ((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) → (𝑓 Fn 𝑥 ↔ 𝐻 Fn 𝑧))
2		simpll 518	. . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐻)
3		simpr 109	. . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
4	2, 3	fveq12d 5428	. . . . . 6 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓‘𝑦) = (𝐻‘𝑤))
5	2, 3	reseq12d 4820	. . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓 ↾ 𝑦) = (𝐻 ↾ 𝑤))
6	5	fveq2d 5425	. . . . . 6 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(𝐻 ↾ 𝑤)))
7	4, 6	eqeq12d 2154	. . . . 5 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤))))
8		simplr 519	. . . . 5 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
9	7, 8	cbvraldva2 2661	. . . 4 ⊢ ((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤))))
10	1, 9	anbi12d 464	. . 3 ⊢ ((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ (𝐻 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤)))))
11	10	cbvrexdva 2664	. 2 ⊢ (𝑓 = 𝐻 → (∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ ∃𝑧 ∈ 𝑋 (𝐻 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤)))))
12		tfr1onlem3ag.1	. 2 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
13	11, 12	elab2g 2831	1 ⊢ (𝐻 ∈ 𝑉 → (𝐻 ∈ 𝐴 ↔ ∃𝑧 ∈ 𝑋 (𝐻 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤)))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417   ↾ cres 4541   Fn wfn 5118  ‘cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131

This theorem is referenced by:  tfr1onlem3  6235  tfr1onlemsucaccv  6238  tfr1onlembxssdm  6240  tfr1onlemres  6246