Theorem fniniseg2 5618

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 5617	. 2 |- ( F Fn A -> ( `' F " { B } ) = { x e. A | ( F ` x ) e. { B } } )
2		funfvex 5513	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. _V )
3		elsng 3598	. . . . 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 5298	. . 3 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) e. { B } <-> ( F ` x ) = B ) )
6	5	rabbidva 2718	. 2 |- ( F Fn A -> { x e. A | ( F ` x ) e. { B } } = { x e. A | ( F ` x ) = B } )
7	1, 6	eqtrd 2203	1 |- ( F Fn A -> ( `' F " { B } ) = { x e. A | ( F ` x ) = B } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   {crab 2452   _Vcvv 2730   {csn 3583   `'ccnv 4610   dom cdm 4611   "cima 4614   Fun wfun 5192   Fn wfn 5193   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by: (None)