Theorem funimacnv 5423

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 4753	. . 3 |- ( F " ( `' F " A ) ) = ran ( F |` ( `' F " A ) )
2		funcnvres2 5422	. . . 4 |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )
3	2	rneqd 4977	. . 3 |- ( Fun F -> ran `' ( `' F |` A ) = ran ( F |` ( `' F " A ) ) )
4	1, 3	eqtr4id 2284	. 2 |- ( Fun F -> ( F " ( `' F " A ) ) = ran `' ( `' F |` A ) )
5		df-rn 4751	. . . 4 |- ran F = dom `' F
6	5	ineq2i 3416	. . 3 |- ( A i^i ran F ) = ( A i^i dom `' F )
7		dmres 5050	. . 3 |- dom ( `' F |` A ) = ( A i^i dom `' F )
8		dfdm4 4939	. . 3 |- dom ( `' F |` A ) = ran `' ( `' F |` A )
9	6, 7, 8	3eqtr2ri 2260	. 2 |- ran `' ( `' F |` A ) = ( A i^i ran F )
10	4, 9	eqtrdi 2281	1 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )

Syntax hints:   -> wi 4   = wceq 1398   i^i cin 3209   `'ccnv 4739   dom cdm 4740   ran crn 4741   |` cres 4742   "cima 4743   Fun wfun 5337

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 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-br 4103  df-opab 4165  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-fun 5345

This theorem is referenced by:  funimass1  5424  funimass2  5425