Theorem fimadmfo 5622

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 5537	. 2 |- ( F : A --> B -> dom F = A )
2		ffn 5531	. . . . 5 |- ( F : A --> B -> F Fn A )
3	2	adantr 276	. . . 4 |- ( ( F : A --> B /\ dom F = A ) -> F Fn A )
4		dffn4 5619	. . . 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 5120	. . . . . . 7 |- ( A = dom F -> ( F " A ) = ( F " dom F ) )
7		imadmrn 5134	. . . . . . 7 |- ( F " dom F ) = ran F
8	6, 7	eqtrdi 2287	. . . . . 6 |- ( A = dom F -> ( F " A ) = ran F )
9	8	eqcoms 2241	. . . . 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 5611	. . . 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 425	1 |- ( F : A --> B -> F : A -onto-> ( F " A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   dom cdm 4772   ran crn 4773   "cima 4775   Fn wfn 5370   -->wf 5371   -onto->wfo 5373

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-xp 4778  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-fn 5378  df-f 5379  df-fo 5381

This theorem is referenced by:  wrdsymb  11315