Theorem tfrlem4 7067

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:   ,,,,   ,

Proof of Theorem tfrlem4

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . 4
2	1	tfrlem3 7066	. . 3
3	2	abeq2i 2584	. 2
4		fnfun 5683	. . . 4
5	4	adantr 465	. . 3
6	5	rexlimivw 2946	. 2
7	3, 6	sylbi 195	1

Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  E.wrex 2808  con0 4883  |`cres 5006  Funwfun 5587  Fnwfn 5588  `cfv 5593

This theorem is referenced by:  tfrlem6  7070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-res 5016  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601