Theorem tfrlem3 5848

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 2538	. . 3 ⊢ g ∈ V
3	1, 2	tfrlem3a 5847	. 2 ⊢ (g ∈ A ↔ ∃z ∈ On (g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w))))
4	3	abbi2i 2134	1 ⊢ A = {g ∣ ∃z ∈ On (g Fn z ∧ ∀w ∈ z (g‘w) = (𝐹‘(g ↾ w)))}

Syntax hints:   ∧ wa 97   = wceq 1228  {cab 2008  ∀wral 2284  ∃wrex 2285  Oncon0 4049   ↾ cres 4274   Fn wfn 4824  ‘cfv 4829

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-res 4284  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837

This theorem is referenced by:  tfrlem4  5851  tfrlem8  5856  tfrlemi1  5867  tfrexlem  5870  tfri1d  5871  tfri1  5873  tfrex  5876