Theorem tfrlem3 5867

Description: Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.)

Hypothesis

Ref	Expression
tfrlem3.1	⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}

Assertion

Ref	Expression
tfrlem3	⊢ A = {g ∣ ∃z ∈ On (g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w)))}

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

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

Proof of Theorem tfrlem3

Step	Hyp	Ref	Expression
1		tfrlem3.1	. . 3 ⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
2		vex 2554	. . 3 ⊢ g ∈ V
3	1, 2	tfrlem3a 5866	. 2 ⊢ (g ∈ A ↔ ∃z ∈ On (g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))))
4	3	abbi2i 2149	1 ⊢ A = {g ∣ ∃z ∈ On (g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w)))}

Syntax hints:   ∧ wa 97   = wceq 1242  {cab 2023  ∀wral 2300  ∃wrex 2301  Oncon0 4066   ↾ cres 4290   Fn wfn 4840  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by:  tfrlem4  5870  tfrlem8  5875  tfrlemi1  5887  tfrexlem  5889  tfri1d  5890  tfrex  5895