Theorem foima2 5801

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 5488). (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 5488	. . . 4 |- ( F : A -onto-> B -> ( F " A ) = B )
2	1	eqcomd 2202	. . 3 |- ( F : A -onto-> B -> B = ( F " A ) )
3	2	eleq2d 2266	. 2 |- ( F : A -onto-> B -> ( Y e. B <-> Y e. ( F " A ) ) )
4		fofn 5485	. . 3 |- ( F : A -onto-> B -> F Fn A )
5		ssid 3204	. . 3 |- A C_ A
6		fvelimab 5620	. . . 4 |- ( ( F Fn A /\ A C_ A ) -> ( Y e. ( F " A ) <-> E. x e. A ( F ` x ) = Y ) )
7		eqcom 2198	. . . . 5 |- ( ( F ` x ) = Y <-> Y = ( F ` x ) )
8	7	rexbii 2504	. . . 4 |- ( E. x e. A ( F ` x ) = Y <-> E. x e. A Y = ( F ` x ) )
9	6, 8	bitrdi 196	. . 3 |- ( ( F Fn A /\ A C_ A ) -> ( Y e. ( F " A ) <-> E. x e. A Y = ( F ` x ) ) )
10	4, 5, 9	sylancl 413	. 2 |- ( F : A -onto-> B -> ( Y e. ( F " A ) <-> E. x e. A Y = ( F ` x ) ) )
11	3, 10	bitrd 188	1 |- ( F : A -onto-> B -> ( Y e. B <-> E. x e. A Y = ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   E.wrex 2476   C_ wss 3157   "cima 4667   Fn wfn 5254   -onto->wfo 5257   ` cfv 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fo 5265  df-fv 5267

This theorem is referenced by:  foelrn  5802