Theorem tfr1onlem3ag 6362

Description: Lemma for transfinite recursion. This lemma changes some bound variables in 𝐴 (version of tfrlem3ag 6334 but for tfr1on 6375 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 5328	. . . 4 ⊢ ((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) → (𝑓 Fn 𝑥 ↔ 𝐻 Fn 𝑧))
2		simpll 527	. . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐻)
3		simpr 110	. . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
4	2, 3	fveq12d 5541	. . . . . 6 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓‘𝑦) = (𝐻‘𝑤))
5	2, 3	reseq12d 4926	. . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓 ↾ 𝑦) = (𝐻 ↾ 𝑤))
6	5	fveq2d 5538	. . . . . 6 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(𝐻 ↾ 𝑤)))
7	4, 6	eqeq12d 2204	. . . . 5 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤))))
8		simplr 528	. . . . 5 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
9	7, 8	cbvraldva2 2725	. . . 4 ⊢ ((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤))))
10	1, 9	anbi12d 473	. . 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 2728	. 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 2899	1 ⊢ (𝐻 ∈ 𝑉 → (𝐻 ∈ 𝐴 ↔ ∃𝑧 ∈ 𝑋 (𝐻 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤)))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2160  {cab 2175  ∀wral 2468  ∃wrex 2469   ↾ cres 4646   Fn wfn 5230  ‘cfv 5235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243

This theorem is referenced by:  tfr1onlem3  6363  tfr1onlemsucaccv  6366  tfr1onlembxssdm  6368  tfr1onlemres  6374