Theorem fimadmfo 5577

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 5495	. 2 |- ( F : A --> B -> dom F = A )
2		ffn 5489	. . . . 5 |- ( F : A --> B -> F Fn A )
3	2	adantr 276	. . . 4 |- ( ( F : A --> B /\ dom F = A ) -> F Fn A )
4		dffn4 5574	. . . 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 5078	. . . . . . 7 |- ( A = dom F -> ( F " A ) = ( F " dom F ) )
7		imadmrn 5092	. . . . . . 7 |- ( F " dom F ) = ran F
8	6, 7	eqtrdi 2280	. . . . . 6 |- ( A = dom F -> ( F " A ) = ran F )
9	8	eqcoms 2234	. . . . 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 5566	. . . 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 1398   dom cdm 4731   ran crn 4732   "cima 4734   Fn wfn 5328   -->wf 5329   -onto->wfo 5331

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-fn 5336  df-f 5337  df-fo 5339

This theorem is referenced by:  wrdsymb  11188