Theorem foima 5526

Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)

Assertion

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

Proof of Theorem foima

Step	Hyp	Ref	Expression
1		imadmrn 5052	. 2 |- ( F " dom F ) = ran F
2		fof 5521	. . . 4 |- ( F : A -onto-> B -> F : A --> B )
3		fdm 5452	. . . 4 |- ( F : A --> B -> dom F = A )
4	2, 3	syl 14	. . 3 |- ( F : A -onto-> B -> dom F = A )
5	4	imaeq2d 5042	. 2 |- ( F : A -onto-> B -> ( F " dom F ) = ( F " A ) )
6		forn 5524	. 2 |- ( F : A -onto-> B -> ran F = B )
7	1, 5, 6	3eqtr3a 2264	1 |- ( F : A -onto-> B -> ( F " A ) = B )

Syntax hints:   -> wi 4   = wceq 1373   dom cdm 4694   ran crn 4695   "cima 4697   -->wf 5287   -onto->wfo 5289

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 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2779  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-br 4061  df-opab 4123  df-xp 4700  df-cnv 4702  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-fn 5294  df-f 5295  df-fo 5297

This theorem is referenced by:  foimacnv  5563  foima2  5845  fiintim  7056  fidcenumlemr  7085