Theorem tfr1onlem3 6424

Description: Lemma for transfinite recursion. This lemma changes some bound variables in 𝐴 (version of tfrlem3 6397 but for tfr1on 6436 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 2775	. . 3 ⊢ 𝑔 ∈ V
2		tfr1onlem3ag.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
3	2	tfr1onlem3ag 6423	. . 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 2320	1 ⊢ 𝐴 = {𝑔 ∣ ∃𝑧 ∈ 𝑋 (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))}

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2176  {cab 2191  ∀wral 2484  ∃wrex 2485  Vcvv 2772   ↾ cres 4677   Fn wfn 5266  ‘cfv 5271

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279

This theorem is referenced by:  tfr1on  6436