Theorem fimacnvinrn2 6349

Description: Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.)

Assertion

Ref	Expression
fimacnvinrn2	|- ( ( Fun F /\ ran F C_ B ) -> ( `' F " A ) = ( `' F " ( A i^i B ) ) )

Proof of Theorem fimacnvinrn2

Step	Hyp	Ref	Expression
1		inass 3823	. . . 4 |- ( ( A i^i B ) i^i ran F ) = ( A i^i ( B i^i ran F ) )
2		sseqin2 3817	. . . . . . 7 |- ( ran F C_ B <-> ( B i^i ran F ) = ran F )
3	2	biimpi 206	. . . . . 6 |- ( ran F C_ B -> ( B i^i ran F ) = ran F )
4	3	adantl 482	. . . . 5 |- ( ( Fun F /\ ran F C_ B ) -> ( B i^i ran F ) = ran F )
5	4	ineq2d 3814	. . . 4 |- ( ( Fun F /\ ran F C_ B ) -> ( A i^i ( B i^i ran F ) ) = ( A i^i ran F ) )
6	1, 5	syl5eq 2668	. . 3 |- ( ( Fun F /\ ran F C_ B ) -> ( ( A i^i B ) i^i ran F ) = ( A i^i ran F ) )
7	6	imaeq2d 5466	. 2 |- ( ( Fun F /\ ran F C_ B ) -> ( `' F " ( ( A i^i B ) i^i ran F ) ) = ( `' F " ( A i^i ran F ) ) )
8		fimacnvinrn 6348	. . 3 |- ( Fun F -> ( `' F " ( A i^i B ) ) = ( `' F " ( ( A i^i B ) i^i ran F ) ) )
9	8	adantr 481	. 2 |- ( ( Fun F /\ ran F C_ B ) -> ( `' F " ( A i^i B ) ) = ( `' F " ( ( A i^i B ) i^i ran F ) ) )
10		fimacnvinrn 6348	. . 3 |- ( Fun F -> ( `' F " A ) = ( `' F " ( A i^i ran F ) ) )
11	10	adantr 481	. 2 |- ( ( Fun F /\ ran F C_ B ) -> ( `' F " A ) = ( `' F " ( A i^i ran F ) ) )
12	7, 9, 11	3eqtr4rd 2667	1 |- ( ( Fun F /\ ran F C_ B ) -> ( `' F " A ) = ( `' F " ( A i^i B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   i^i cin 3573   C_ wss 3574   `'ccnv 5113   ran crn 5115   "cima 5117   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896

This theorem is referenced by:  eulerpartgbij  30434  orvcval4  30522  preimaioomnf  40929