Theorem foima2 5875

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 5553). (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 5553	. . . 4 |- ( F : A -onto-> B -> ( F " A ) = B )
2	1	eqcomd 2235	. . 3 |- ( F : A -onto-> B -> B = ( F " A ) )
3	2	eleq2d 2299	. 2 |- ( F : A -onto-> B -> ( Y e. B <-> Y e. ( F " A ) ) )
4		fofn 5550	. . 3 |- ( F : A -onto-> B -> F Fn A )
5		ssid 3244	. . 3 |- A C_ A
6		fvelimab 5690	. . . 4 |- ( ( F Fn A /\ A C_ A ) -> ( Y e. ( F " A ) <-> E. x e. A ( F ` x ) = Y ) )
7		eqcom 2231	. . . . 5 |- ( ( F ` x ) = Y <-> Y = ( F ` x ) )
8	7	rexbii 2537	. . . 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 1395   e. wcel 2200   E.wrex 2509   C_ wss 3197   "cima 4722   Fn wfn 5313   -onto->wfo 5316   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326

This theorem is referenced by:  foelrn  5876