Theorem fniniseg2 5289

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 5288	. 2 |- ( F Fn A -> ( `' F " { B } ) = { x e. A | ( F ` x ) e. { B } } )
2		funfvex 5192	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
3		elsng 3390	. . . . 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 4999	. . 3 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) e. { B } <-> ( F ` x ) = B ) )
6	5	rabbidva 2548	. 2 |- ( F Fn A -> { x e. A | ( F ` x ) e. { B } } = { x e. A | ( F ` x ) = B } )
7	1, 6	eqtrd 2072	1 |- ( F Fn A -> ( `' F " { B } ) = { x e. A | ( F ` x ) = B } )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   {crab 2310   _Vcvv 2557   {csn 3375   `'ccnv 4344   dom cdm 4345   "cima 4348   Fun wfun 4896   Fn wfn 4897   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910

This theorem is referenced by: (None)