Theorem fimarab 29445

Description: Expressing the image of a set as a restricted abstract builder. (Contributed by Thierry Arnoux, 27-Jan-2020.)

Assertion

Ref	Expression
fimarab	|- ( ( F : A --> B /\ X C_ A ) -> ( F " X ) = { y e. B | E. x e. X ( F ` x ) = y } )

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

Allowed substitution hints:   A( x)   B( x)

Proof of Theorem fimarab

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ y ( F : A --> B /\ X C_ A )
2		nfcv 2764	. 2 |- F/_ y ( F " X )
3		nfrab1 3122	. 2 |- F/_ y { y e. B | E. x e. X ( F ` x ) = y }
4		ffn 6045	. . . 4 |- ( F : A --> B -> F Fn A )
5		fvelimab 6253	. . . . 5 |- ( ( F Fn A /\ X C_ A ) -> ( y e. ( F " X ) <-> E. x e. X ( F ` x ) = y ) )
6	5	anbi2d 740	. . . 4 |- ( ( F Fn A /\ X C_ A ) -> ( ( y e. B /\ y e. ( F " X ) ) <-> ( y e. B /\ E. x e. X ( F ` x ) = y ) ) )
7	4, 6	sylan 488	. . 3 |- ( ( F : A --> B /\ X C_ A ) -> ( ( y e. B /\ y e. ( F " X ) ) <-> ( y e. B /\ E. x e. X ( F ` x ) = y ) ) )
8		imassrn 5477	. . . . . . 7 |- ( F " X ) C_ ran F
9		frn 6053	. . . . . . 7 |- ( F : A --> B -> ran F C_ B )
10	8, 9	syl5ss 3614	. . . . . 6 |- ( F : A --> B -> ( F " X ) C_ B )
11	10	adantr 481	. . . . 5 |- ( ( F : A --> B /\ X C_ A ) -> ( F " X ) C_ B )
12	11	sseld 3602	. . . 4 |- ( ( F : A --> B /\ X C_ A ) -> ( y e. ( F " X ) -> y e. B ) )
13	12	pm4.71rd 667	. . 3 |- ( ( F : A --> B /\ X C_ A ) -> ( y e. ( F " X ) <-> ( y e. B /\ y e. ( F " X ) ) ) )
14		rabid 3116	. . . 4 |- ( y e. { y e. B | E. x e. X ( F ` x ) = y } <-> ( y e. B /\ E. x e. X ( F ` x ) = y ) )
15	14	a1i 11	. . 3 |- ( ( F : A --> B /\ X C_ A ) -> ( y e. { y e. B | E. x e. X ( F ` x ) = y } <-> ( y e. B /\ E. x e. X ( F ` x ) = y ) ) )
16	7, 13, 15	3bitr4d 300	. 2 |- ( ( F : A --> B /\ X C_ A ) -> ( y e. ( F " X ) <-> y e. { y e. B | E. x e. X ( F ` x ) = y } ) )
17	1, 2, 3, 16	eqrd 3622	1 |- ( ( F : A --> B /\ X C_ A ) -> ( F " X ) = { y e. B | E. x e. X ( F ` x ) = y } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   C_ wss 3574   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  locfinreflem  29907