Theorem fimadmfo 5518

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 5440	. 2 |- ( F : A --> B -> dom F = A )
2		ffn 5434	. . . . 5 |- ( F : A --> B -> F Fn A )
3	2	adantr 276	. . . 4 |- ( ( F : A --> B /\ dom F = A ) -> F Fn A )
4		dffn4 5515	. . . 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 5026	. . . . . . 7 |- ( A = dom F -> ( F " A ) = ( F " dom F ) )
7		imadmrn 5040	. . . . . . 7 |- ( F " dom F ) = ran F
8	6, 7	eqtrdi 2255	. . . . . 6 |- ( A = dom F -> ( F " A ) = ran F )
9	8	eqcoms 2209	. . . . 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 5507	. . . 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 1373   dom cdm 4682   ran crn 4683   "cima 4685   Fn wfn 5274   -->wf 5275   -onto->wfo 5277

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-br 4051  df-opab 4113  df-xp 4688  df-cnv 4690  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-fn 5282  df-f 5283  df-fo 5285

This theorem is referenced by:  wrdsymb  11038