Theorem tfr1onlem3ag 6581

Description: Lemma for transfinite recursion. This lemma changes some bound variables in 𝐴 (version of tfrlem3ag 6553 but for tfr1on 6594 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 5454	. . . 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 5682	. . . . . 6 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓‘𝑦) = (𝐻‘𝑤))
5	2, 3	reseq12d 5044	. . . . . . 7 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓 ↾ 𝑦) = (𝐻 ↾ 𝑤))
6	5	fveq2d 5679	. . . . . 6 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(𝐻 ↾ 𝑤)))
7	4, 6	eqeq12d 2249	. . . . 5 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (𝐻‘𝑤) = (𝐺‘(𝐻 ↾ 𝑤))))
8		simplr 529	. . . . 5 ⊢ (((𝑓 = 𝐻 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
9	7, 8	cbvraldva2 2787	. . . 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 2790	. 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 2967	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 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523   ↾ cres 4756   Fn wfn 5352  ‘cfv 5357

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365

This theorem is referenced by:  tfr1onlem3  6582  tfr1onlemsucaccv  6585  tfr1onlembxssdm  6587  tfr1onlemres  6593