Theorem funimacnv 5369

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 4706	. . 3 |- ( F " ( `' F " A ) ) = ran ( F |` ( `' F " A ) )
2		funcnvres2 5368	. . . 4 |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )
3	2	rneqd 4926	. . 3 |- ( Fun F -> ran `' ( `' F |` A ) = ran ( F |` ( `' F " A ) ) )
4	1, 3	eqtr4id 2259	. 2 |- ( Fun F -> ( F " ( `' F " A ) ) = ran `' ( `' F |` A ) )
5		df-rn 4704	. . . 4 |- ran F = dom `' F
6	5	ineq2i 3379	. . 3 |- ( A i^i ran F ) = ( A i^i dom `' F )
7		dmres 4999	. . 3 |- dom ( `' F |` A ) = ( A i^i dom `' F )
8		dfdm4 4889	. . 3 |- dom ( `' F |` A ) = ran `' ( `' F |` A )
9	6, 7, 8	3eqtr2ri 2235	. 2 |- ran `' ( `' F |` A ) = ( A i^i ran F )
10	4, 9	eqtrdi 2256	1 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )

Syntax hints:   -> wi 4   = wceq 1373   i^i cin 3173   `'ccnv 4692   dom cdm 4693   ran crn 4694   |` cres 4695   "cima 4696   Fun wfun 5284

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 4178  ax-pow 4234  ax-pr 4269

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 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-fun 5292

This theorem is referenced by:  funimass1  5370  funimass2  5371