Theorem funimacnv 5434

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		df-ima 4764	. . 3 |- ( F " ( `' F " A ) ) = ran ( F |` ( `' F " A ) )
2		funcnvres2 5433	. . . 4 |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )
3	2	rneqd 4988	. . 3 |- ( Fun F -> ran `' ( `' F |` A ) = ran ( F |` ( `' F " A ) ) )
4	1, 3	eqtr4id 2286	. 2 |- ( Fun F -> ( F " ( `' F " A ) ) = ran `' ( `' F |` A ) )
5		df-rn 4762	. . . 4 |- ran F = dom `' F
6	5	ineq2i 3421	. . 3 |- ( A i^i ran F ) = ( A i^i dom `' F )
7		dmres 5061	. . 3 |- dom ( `' F |` A ) = ( A i^i dom `' F )
8		dfdm4 4950	. . 3 |- dom ( `' F |` A ) = ran `' ( `' F |` A )
9	6, 7, 8	3eqtr2ri 2262	. 2 |- ran `' ( `' F |` A ) = ( A i^i ran F )
10	4, 9	eqtrdi 2283	1 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )

Syntax hints:   -> wi 4   = wceq 1398   i^i cin 3212   `'ccnv 4750   dom cdm 4751   ran crn 4752   |` cres 4753   "cima 4754   Fun wfun 5348

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-fun 5356

This theorem is referenced by:  funimass1  5435  funimass2  5436