Theorem funfv2f 6267

Description: The value of a function. Version of funfv2 6266 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)

Hypotheses

Ref	Expression
funfv2f.1	|- F/_ y A
funfv2f.2	|- F/_ y F

Assertion

Ref	Expression
funfv2f	|- ( Fun F -> ( F ` A ) = U. { y | A F y } )

Proof of Theorem funfv2f

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfv2 6266	. 2 |- ( Fun F -> ( F ` A ) = U. { w | A F w } )
2		funfv2f.1	. . . . 5 |- F/_ y A
3		funfv2f.2	. . . . 5 |- F/_ y F
4		nfcv 2764	. . . . 5 |- F/_ y w
5	2, 3, 4	nfbr 4699	. . . 4 |- F/ y A F w
6		nfv 1843	. . . 4 |- F/ w A F y
7		breq2 4657	. . . 4 |- ( w = y -> ( A F w <-> A F y ) )
8	5, 6, 7	cbvab 2746	. . 3 |- { w | A F w } = { y | A F y }
9	8	unieqi 4445	. 2 |- U. { w | A F w } = U. { y | A F y }
10	1, 9	syl6eq 2672	1 |- ( Fun F -> ( F ` A ) = U. { y | A F y } )

Syntax hints:   -> wi 4   = wceq 1483   {cab 2608   F/_wnfc 2751   U.cuni 4436   class class class wbr 4653   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by: (None)