Theorem fnniniseg2 5726

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 5724	. 2 |- ( F Fn A -> ( `' F " ( _V \ { B } ) ) = { x e. A | ( F ` x ) e. ( _V \ { B } ) } )
2		eldifsn 3771	. . . 4 |- ( ( F ` x ) e. ( _V \ { B } ) <-> ( ( F ` x ) e. _V /\ ( F ` x ) =/= B ) )
3		funfvex 5616	. . . . . 6 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
4	3	funfni 5395	. . . . 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 2764	. 2 |- ( F Fn A -> { x e. A | ( F ` x ) e. ( _V \ { B } ) } = { x e. A | ( F ` x ) =/= B } )
8	1, 7	eqtrd 2240	1 |- ( F Fn A -> ( `' F " ( _V \ { B } ) ) = { x e. A | ( F ` x ) =/= B } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   =/= wne 2378   {crab 2490   _Vcvv 2776   \ cdif 3171   {csn 3643   `'ccnv 4692   "cima 4696   Fn wfn 5285   ` cfv 5290

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by: (None)