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Theorem tfrlem4 6176
 Description: Lemma for transfinite recursion. is the class of all "acceptable" functions, and 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
Assertion
Ref Expression
tfrlem4
Distinct variable groups:   ,,,,   ,
Allowed substitution hints:   (,,)

Proof of Theorem tfrlem4
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlem.1 . . . 4
21tfrlem3 6174 . . 3
32abeq2i 2226 . 2
4 fnfun 5188 . . . 4
65rexlimivw 2520 . 2
73, 6sylbi 120 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1314   wcel 1463  cab 2101  wral 2391  wrex 2392  con0 4253   cres 4509   wfun 5085   wfn 5086  cfv 5091 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-res 4519  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099 This theorem is referenced by:  tfrlem6  6179
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