Theorem fnniniseg2 5551

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 5549	. 2 |- ( F Fn A -> ( `' F " ( _V \ { B } ) ) = { x e. A | ( F ` x ) e. ( _V \ { B } ) } )
2		eldifsn 3658	. . . 4 |- ( ( F ` x ) e. ( _V \ { B } ) <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= B ) )
3		funfvex 5446	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
4	3	funfni 5231	. . . . 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 2677	. 2 |- ( F Fn A -> { x e. A | ( F ` x ) e. ( _V \ { B } ) } = { x e. A | ( F ` x ) =/= B } )
8	1, 7	eqtrd 2173	1 |- ( F Fn A -> ( `' F " ( _V \ { B } ) ) = { x e. A | ( F ` x ) =/= B } )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   =/= wne 2309   {crab 2421   _Vcvv 2689   \ cdif 3073   {csn 3532   `'ccnv 4546   "cima 4550   Fn wfn 5126   ` cfv 5131

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139

This theorem is referenced by: (None)