Theorem tfr1onlem3ag 6481

Description: Lemma for transfinite recursion. This lemma changes some bound variables in 𝐴 (version of tfrlem3ag 6453 but for tfr1on 6494 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 5413	. . . 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 5633	. . . . . 6 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓‘𝑦) = (𝐻‘𝑤))
5	2, 3	reseq12d 5005	. . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓 ↾ 𝑦) = (𝐻 ↾ 𝑤))
6	5	fveq2d 5630	. . . . . 6 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(𝐻 ↾ 𝑤)))
7	4, 6	eqeq12d 2244	. . . . 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 2772	. . . 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 2775	. 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 2950	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 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509   ↾ cres 4720   Fn wfn 5312  ‘cfv 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325

This theorem is referenced by:  tfr1onlem3  6482  tfr1onlemsucaccv  6485  tfr1onlembxssdm  6487  tfr1onlemres  6493