Theorem fniniseg2 5607

Description: Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
fniniseg2	|- ( F Fn A -> ( `' F " { B } ) = { x e. A | ( F ` x ) = B } )

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

Proof of Theorem fniniseg2

Step	Hyp	Ref	Expression
1		fncnvima2 5606	. 2 |- ( F Fn A -> ( `' F " { B } ) = { x e. A | ( F ` x ) e. { B } } )
2		funfvex 5503	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
3		elsng 3591	. . . . 5 |- ( ( F ` x ) e. _V -> ( ( F ` x ) e. { B } <-> ( F ` x ) = B ) )
4	2, 3	syl 14	. . . 4 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) e. { B } <-> ( F ` x ) = B ) )
5	4	funfni 5288	. . 3 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) e. { B } <-> ( F ` x ) = B ) )
6	5	rabbidva 2714	. 2 |- ( F Fn A -> { x e. A | ( F ` x ) e. { B } } = { x e. A | ( F ` x ) = B } )
7	1, 6	eqtrd 2198	1 |- ( F Fn A -> ( `' F " { B } ) = { x e. A | ( F ` x ) = B } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {crab 2448   _Vcvv 2726   {csn 3576   `'ccnv 4603   dom cdm 4604   "cima 4607   Fun wfun 5182   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-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-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  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)