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Theorem tfrlem4 5851
Description: Lemma for transfinite recursion. A is the class of all "acceptable" functions, and 𝐹 is their union. First we show that an acceptable function is in fact a function. (Contributed by NM, 9-Apr-1995.)
Hypothesis
Ref Expression
tfrlem.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
Assertion
Ref Expression
tfrlem4 (g A → Fun g)
Distinct variable groups:   f,g,x,y,𝐹   A,g
Allowed substitution hints:   A(x,y,f)

Proof of Theorem tfrlem4
Dummy variables z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . 4 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
21tfrlem3 5848 . . 3 A = {gz On (g Fn z w z (gw) = (𝐹‘(gw)))}
32abeq2i 2130 . 2 (g Az On (g Fn z w z (gw) = (𝐹‘(gw))))
4 fnfun 4922 . . . 4 (g Fn z → Fun g)
54adantr 261 . . 3 ((g Fn z w z (gw) = (𝐹‘(gw))) → Fun g)
65rexlimivw 2407 . 2 (z On (g Fn z w z (gw) = (𝐹‘(gw))) → Fun g)
73, 6sylbi 114 1 (g A → Fun g)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1228   wcel 1374  {cab 2008  wral 2284  wrex 2285  Oncon0 4049  cres 4274  Fun wfun 4823   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:  tfrlem6  5854
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