Theorem fnniniseg2 5608

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 5606	. 2 |- ( F Fn A -> ( `' F " ( _V \ { B } ) ) = { x e. A | ( F ` x ) e. ( _V \ { B } ) } )
2		eldifsn 3703	. . . 4 |- ( ( F ` x ) e. ( _V \ { B } ) <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= B ) )
3		funfvex 5503	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
4	3	funfni 5288	. . . . 5 |- ( ( F Fn A /\ x e. A ) -> ( F ` x ) e. _V )
5	4	biantrurd 303	. . . 4 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) =/= B <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= B ) ) )
6	2, 5	bitr4id 198	. . 3 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) e. ( _V \ { B } ) <-> ( F ` x ) =/= B ) )
7	6	rabbidva 2714	. 2 |- ( F Fn A -> { x e. A | ( F ` x ) e. ( _V \ { B } ) } = { x e. A | ( F ` x ) =/= B } )
8	1, 7	eqtrd 2198	1 |- ( F Fn A -> ( `' F " ( _V \ { B } ) ) = { x e. A | ( F ` x ) =/= B } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   =/= wne 2336   {crab 2448   _Vcvv 2726   \ cdif 3113   {csn 3576   `'ccnv 4603   "cima 4607   Fn wfn 5183   ` cfv 5188

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-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-fv 5196

This theorem is referenced by: (None)