Theorem funimacnv 5350

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 4688	. . 3 |- ( F " ( `' F " A ) ) = ran ( F |` ( `' F " A ) )
2		funcnvres2 5349	. . . 4 |- ( Fun F -> `' ( `' F |` A ) = ( F |` ( `' F " A ) ) )
3	2	rneqd 4907	. . 3 |- ( Fun F -> ran `' ( `' F |` A ) = ran ( F |` ( `' F " A ) ) )
4	1, 3	eqtr4id 2257	. 2 |- ( Fun F -> ( F " ( `' F " A ) ) = ran `' ( `' F |` A ) )
5		df-rn 4686	. . . 4 |- ran F = dom `' F
6	5	ineq2i 3371	. . 3 |- ( A i^i ran F ) = ( A i^i dom `' F )
7		dmres 4980	. . 3 |- dom ( `' F |` A ) = ( A i^i dom `' F )
8		dfdm4 4870	. . 3 |- dom ( `' F |` A ) = ran `' ( `' F |` A )
9	6, 7, 8	3eqtr2ri 2233	. 2 |- ran `' ( `' F |` A ) = ( A i^i ran F )
10	4, 9	eqtrdi 2254	1 |- ( Fun F -> ( F " ( `' F " A ) ) = ( A i^i ran F ) )

Syntax hints:   -> wi 4   = wceq 1373   i^i cin 3165   `'ccnv 4674   dom cdm 4675   ran crn 4676   |` cres 4677   "cima 4678   Fun wfun 5265

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-fun 5273

This theorem is referenced by:  funimass1  5351  funimass2  5352