Theorem tfr1onlem3 6306

Description: Lemma for transfinite recursion. This lemma changes some bound variables in 𝐴 (version of tfrlem3 6279 but for tfr1on 6318 related lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.)

Hypothesis

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

Assertion

Ref	Expression
tfr1onlem3	⊢ 𝐴 = {𝑔 ∣ ∃𝑧 ∈ 𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))}

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

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

Proof of Theorem tfr1onlem3

Step	Hyp	Ref	Expression
1		vex 2729	. . 3 ⊢ 𝑔 ∈ V
2		tfr1onlem3ag.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
3	2	tfr1onlem3ag 6305	. . 3 ⊢ (𝑔 ∈ V → (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ 𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))))
4	1, 3	ax-mp 5	. 2 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ 𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
5	4	abbi2i 2281	1 ⊢ 𝐴 = {𝑔 ∣ ∃𝑧 ∈ 𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))}

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  ∃wrex 2445  Vcvv 2726   ↾ cres 4606   Fn wfn 5183  ‘cfv 5188

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by:  tfr1on  6318