Theorem funimacnv 5406

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 4738	. . 3 |- ( F " ( `' F " A ) ) = ran ( F |` ( `' F " A ) )
2		funcnvres2 5405	. . . 4 |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )
3	2	rneqd 4961	. . 3 |- ( Fun F -> ran `' ( `' F |` A ) = ran ( F |` ( `' F " A ) ) )
4	1, 3	eqtr4id 2283	. 2 |- ( Fun F -> ( F " ( `' F " A ) ) = ran `' ( `' F |` A ) )
5		df-rn 4736	. . . 4 |- ran F = dom `' F
6	5	ineq2i 3405	. . 3 |- ( A i^i ran F ) = ( A i^i dom `' F )
7		dmres 5034	. . 3 |- dom ( `' F |` A ) = ( A i^i dom `' F )
8		dfdm4 4923	. . 3 |- dom ( `' F |` A ) = ran `' ( `' F |` A )
9	6, 7, 8	3eqtr2ri 2259	. 2 |- ran `' ( `' F |` A ) = ( A i^i ran F )
10	4, 9	eqtrdi 2280	1 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )

Syntax hints:   -> wi 4   = wceq 1397   i^i cin 3199   `'ccnv 4724   dom cdm 4725   ran crn 4726   |` cres 4727   "cima 4728   Fun wfun 5320

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328

This theorem is referenced by:  funimass1  5407  funimass2  5408