Theorem foima2 5530

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 5238). (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 5238	. . . 4 |- ( F : A -onto-> B -> ( F " A ) = B )
2	1	eqcomd 2093	. . 3 |- ( F : A -onto-> B -> B = ( F " A ) )
3	2	eleq2d 2157	. 2 |- ( F : A -onto-> B -> ( Y e. B <-> Y e. ( F " A ) ) )
4		fofn 5235	. . 3 |- ( F : A -onto-> B -> F Fn A )
5		ssid 3044	. . 3 |- A C_ A
6		fvelimab 5360	. . . 4 |- ( ( F Fn A /\ A C_ A ) -> ( Y e. ( F " A ) <-> E. x e. A ( F ` x ) = Y ) )
7		eqcom 2090	. . . . 5 |- ( ( F ` x ) = Y <-> Y = ( F ` x ) )
8	7	rexbii 2385	. . . 4 |- ( E. x e. A ( F ` x ) = Y <-> E. x e. A Y = ( F ` x ) )
9	6, 8	syl6bb 194	. . 3 |- ( ( F Fn A /\ A C_ A ) -> ( Y e. ( F " A ) <-> E. x e. A Y = ( F ` x ) ) )
10	4, 5, 9	sylancl 404	. 2 |- ( F : A -onto-> B -> ( Y e. ( F " A ) <-> E. x e. A Y = ( F ` x ) ) )
11	3, 10	bitrd 186	1 |- ( F : A -onto-> B -> ( Y e. B <-> E. x e. A Y = ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   E.wrex 2360   C_ wss 2999   "cima 4441   Fn wfn 5010   -onto->wfo 5013   ` cfv 5015

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023

This theorem is referenced by:  foelrn  5531