Theorem fimadmfo 5477

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 5401	. 2 |- ( F : A --> B -> dom F = A )
2		ffn 5395	. . . . 5 |- ( F : A --> B -> F Fn A )
3	2	adantr 276	. . . 4 |- ( ( F : A --> B /\ dom F = A ) -> F Fn A )
4		dffn4 5474	. . . 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 4995	. . . . . . 7 |- ( A = dom F -> ( F " A ) = ( F " dom F ) )
7		imadmrn 5009	. . . . . . 7 |- ( F " dom F ) = ran F
8	6, 7	eqtrdi 2242	. . . . . 6 |- ( A = dom F -> ( F " A ) = ran F )
9	8	eqcoms 2196	. . . . 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 5466	. . . 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 1364   dom cdm 4655   ran crn 4656   "cima 4658   Fn wfn 5241   -->wf 5242   -onto->wfo 5244

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4661  df-cnv 4663  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fn 5249  df-f 5250  df-fo 5252

This theorem is referenced by:  wrdsymb  10931