Theorem fnniniseg2 5803

Description: Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
fnniniseg2	|- ( F Fn A -> ( `' F " ( _V \ { B } ) ) = { x e. A | ( F ` x ) =/= B } )

Distinct variable groups:   x, A   x, F   x, B

Proof of Theorem fnniniseg2

Step	Hyp	Ref	Expression
1		fncnvima2 5801	. 2 |- ( F Fn A -> ( `' F " ( _V \ { B } ) ) = { x e. A | ( F ` x ) e. ( _V \ { B } ) } )
2		eldifsn 3822	. . . 4 |- ( ( F ` x ) e. ( _V \ { B } ) <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= B ) )
3		funfvex 5689	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
4	3	funfni 5460	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
5	4	biantrurd 305	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) =/= B <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= B ) ) )
6	2, 5	bitr4id 199	. . 3 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) e. ( _V \ { B } ) <-> ( F ` x ) =/= B ) )
7	6	rabbidva 2803	. 2 |- ( F Fn A -> { x e. A | ( F ` x ) e. ( _V \ { B } ) } = { x e. A | ( F ` x ) =/= B } )
8	1, 7	eqtrd 2267	1 |- ( F Fn A -> ( `' F " ( _V \ { B } ) ) = { x e. A | ( F ` x ) =/= B } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   =/= wne 2414   {crab 2526   _Vcvv 2815   \ cdif 3210   {csn 3691   `'ccnv 4750   "cima 4754   Fn wfn 5349   ` cfv 5354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362

This theorem is referenced by: (None)