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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 = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
Assertion
Ref Expression
tfrlem3 A = {gz On (g Fn z w z (gw) = (𝐹‘(gw)))}
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
StepHypRef Expression
1 tfrlem3.1 . . 3 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
2 vex 2538 . . 3 g V
31, 2tfrlem3a 5847 . 2 (g Az On (g Fn z w z (gw) = (𝐹‘(gw))))
43abbi2i 2134 1 A = {gz On (g Fn z w z (gw) = (𝐹‘(gw)))}
 Colors of variables: wff set class 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
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