Theorem funimacnv 5194

Description: The image of the preimage of a function. (Contributed by NM, 25-May-2004.)

Assertion

Ref	Expression
funimacnv	|- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )

Proof of Theorem funimacnv

Step	Hyp	Ref	Expression
1		funcnvres2 5193	. . . 4 |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )
2	1	rneqd 4763	. . 3 |- ( Fun F -> ran `' ( `' F |` A ) = ran ( F |` ( `' F " A ) ) )
3		df-ima 4547	. . 3 |- ( F " ( `' F " A ) ) = ran ( F |` ( `' F " A ) )
4	2, 3	syl6reqr 2189	. 2 |- ( Fun F -> ( F " ( `' F " A ) ) = ran `' ( `' F |` A ) )
5		df-rn 4545	. . . 4 |- ran F = dom `' F
6	5	ineq2i 3269	. . 3 |- ( A i^i ran F ) = ( A i^i dom `' F )
7		dmres 4835	. . 3 |- dom ( `' F |` A ) = ( A i^i dom `' F )
8		dfdm4 4726	. . 3 |- dom ( `' F |` A ) = ran `' ( `' F |` A )
9	6, 7, 8	3eqtr2ri 2165	. 2 |- ran `' ( `' F |` A ) = ( A i^i ran F )
10	4, 9	syl6eq 2186	1 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )

Syntax hints:   -> wi 4   = wceq 1331   i^i cin 3065   `'ccnv 4533   dom cdm 4534   ran crn 4535   |` cres 4536   "cima 4537   Fun wfun 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-fun 5120

This theorem is referenced by:  funimass1  5195  funimass2  5196