Theorem fimadmfo 5553

Description: A function is a function onto the image of its domain. (Contributed by AV, 1-Dec-2022.)

Assertion

Ref	Expression
fimadmfo	|- ( F : A --> B -> F : A -onto-> ( F " A ) )

Proof of Theorem fimadmfo

Step	Hyp	Ref	Expression
1		fdm 5475	. 2 |- ( F : A --> B -> dom F = A )
2		ffn 5469	. . . . 5 |- ( F : A --> B -> F Fn A )
3	2	adantr 276	. . . 4 |- ( ( F : A --> B /\ dom F = A ) -> F Fn A )
4		dffn4 5550	. . . 4 |- ( F Fn A <-> F : A -onto-> ran F )
5	3, 4	sylib 122	. . 3 |- ( ( F : A --> B /\ dom F = A ) -> F : A -onto-> ran F )
6		imaeq2 5060	. . . . . . 7 |- ( A = dom F -> ( F " A ) = ( F " dom F ) )
7		imadmrn 5074	. . . . . . 7 |- ( F " dom F ) = ran F
8	6, 7	eqtrdi 2278	. . . . . 6 |- ( A = dom F -> ( F " A ) = ran F )
9	8	eqcoms 2232	. . . . 5 |- ( dom F = A -> ( F " A ) = ran F )
10	9	adantl 277	. . . 4 |- ( ( F : A --> B /\ dom F = A ) -> ( F " A ) = ran F )
11		foeq3 5542	. . . 4 |- ( ( F " A ) = ran F -> ( F : A -onto-> ( F " A ) <-> F : A -onto-> ran F ) )
12	10, 11	syl 14	. . 3 |- ( ( F : A --> B /\ dom F = A ) -> ( F : A -onto-> ( F " A ) <-> F : A -onto-> ran F ) )
13	5, 12	mpbird 167	. 2 |- ( ( F : A --> B /\ dom F = A ) -> F : A -onto-> ( F " A ) )
14	1, 13	mpdan 421	1 |- ( F : A --> B -> F : A -onto-> ( F " A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   dom cdm 4716   ran crn 4717   "cima 4719   Fn wfn 5309   -->wf 5310   -onto->wfo 5312

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 4201  ax-pow 4257  ax-pr 4292

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-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4722  df-cnv 4724  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-fn 5317  df-f 5318  df-fo 5320

This theorem is referenced by:  wrdsymb  11085