Theorem foima2 5795

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 5482). (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 5482	. . . 4 |- ( F : A -onto-> B -> ( F " A ) = B )
2	1	eqcomd 2199	. . 3 |- ( F : A -onto-> B -> B = ( F " A ) )
3	2	eleq2d 2263	. 2 |- ( F : A -onto-> B -> ( Y e. B <-> Y e. ( F " A ) ) )
4		fofn 5479	. . 3 |- ( F : A -onto-> B -> F Fn A )
5		ssid 3200	. . 3 |- A C_ A
6		fvelimab 5614	. . . 4 |- ( ( F Fn A /\ A C_ A ) -> ( Y e. ( F " A ) <-> E. x e. A ( F ` x ) = Y ) )
7		eqcom 2195	. . . . 5 |- ( ( F ` x ) = Y <-> Y = ( F ` x ) )
8	7	rexbii 2501	. . . 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 2164   E.wrex 2473   C_ wss 3154   "cima 4663   Fn wfn 5250   -onto->wfo 5253   ` cfv 5255

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263

This theorem is referenced by:  foelrn  5796