Theorem foima2 5661

Description: Given an onto function, an element is in its codomain if and only if it is the image of an element of its domain (see foima 5358). (Contributed by BJ, 6-Jul-2022.)

Assertion

Ref	Expression
foima2	|- ( F : A -onto-> B -> ( Y e. B <-> E. x e. A Y = ( F ` x ) ) )

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

Allowed substitution hint:   B( x)

Proof of Theorem foima2

Step	Hyp	Ref	Expression
1		foima 5358	. . . 4 |- ( F : A -onto-> B -> ( F " A ) = B )
2	1	eqcomd 2146	. . 3 |- ( F : A -onto-> B -> B = ( F " A ) )
3	2	eleq2d 2210	. 2 |- ( F : A -onto-> B -> ( Y e. B <-> Y e. ( F " A ) ) )
4		fofn 5355	. . 3 |- ( F : A -onto-> B -> F Fn A )
5		ssid 3122	. . 3 |- A C_ A
6		fvelimab 5485	. . . 4 |- ( ( F Fn A /\ A C_ A ) -> ( Y e. ( F " A ) <-> E. x e. A ( F ` x ) = Y ) )
7		eqcom 2142	. . . . 5 |- ( ( F ` x ) = Y <-> Y = ( F ` x ) )
8	7	rexbii 2445	. . . 4 |- ( E. x e. A ( F ` x ) = Y <-> E. x e. A Y = ( F ` x ) )
9	6, 8	syl6bb 195	. . 3 |- ( ( F Fn A /\ A C_ A ) -> ( Y e. ( F " A ) <-> E. x e. A Y = ( F ` x ) ) )
10	4, 5, 9	sylancl 410	. 2 |- ( F : A -onto-> B -> ( Y e. ( F " A ) <-> E. x e. A Y = ( F ` x ) ) )
11	3, 10	bitrd 187	1 |- ( F : A -onto-> B -> ( Y e. B <-> E. x e. A Y = ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   E.wrex 2418   C_ wss 3076   "cima 4550   Fn wfn 5126   -onto->wfo 5129   ` cfv 5131

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139

This theorem is referenced by:  foelrn  5662