Theorem cnvresima 5093

Description: An image under the converse of a restriction. (Contributed by Jeff Hankins, 12-Jul-2009.)

Assertion

Ref	Expression
cnvresima	|- ( `' ( F |` A ) " B ) = ( ( `' F " B ) i^i A )

Proof of Theorem cnvresima

Dummy variables t s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2729	. . . 4 |- t e. _V
2	1	elima3 4953	. . 3 |- ( t e. ( `' ( F |` A ) " B ) <-> E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) )
3	1	elima3 4953	. . . . 5 |- ( t e. ( `' F " B ) <-> E. s ( s e. B /\ <. s , t >. e. `' F ) )
4	3	anbi1i 454	. . . 4 |- ( ( t e. ( `' F " B ) /\ t e. A ) <-> ( E. s ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
5		elin 3305	. . . 4 |- ( t e. ( ( `' F " B ) i^i A ) <-> ( t e. ( `' F " B ) /\ t e. A ) )
6		vex 2729	. . . . . . . . . 10 |- s e. _V
7	6, 1	opelcnv 4786	. . . . . . . . 9 |- ( <. s , t >. e. `' ( F |` A ) <-> <. t , s >. e. ( F |` A ) )
8	6	opelres 4889	. . . . . . . . . 10 |- ( <. t , s >. e. ( F |` A ) <-> ( <. t , s >. e. F /\ t e. A ) )
9	6, 1	opelcnv 4786	. . . . . . . . . . 11 |- ( <. s , t >. e. `' F <-> <. t , s >. e. F )
10	9	anbi1i 454	. . . . . . . . . 10 |- ( ( <. s , t >. e. `' F /\ t e. A ) <-> ( <. t , s >. e. F /\ t e. A ) )
11	8, 10	bitr4i 186	. . . . . . . . 9 |- ( <. t , s >. e. ( F |` A ) <-> ( <. s , t >. e. `' F /\ t e. A ) )
12	7, 11	bitri 183	. . . . . . . 8 |- ( <. s , t >. e. `' ( F |` A ) <-> ( <. s , t >. e. `' F /\ t e. A ) )
13	12	anbi2i 453	. . . . . . 7 |- ( ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> ( s e. B /\ ( <. s , t >. e. `' F /\ t e. A ) ) )
14		anass 399	. . . . . . 7 |- ( ( ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) <-> ( s e. B /\ ( <. s , t >. e. `' F /\ t e. A ) ) )
15	13, 14	bitr4i 186	. . . . . 6 |- ( ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> ( ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
16	15	exbii 1593	. . . . 5 |- ( E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> E. s ( ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
17		19.41v 1890	. . . . 5 |- ( E. s ( ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) <-> ( E. s ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
18	16, 17	bitri 183	. . . 4 |- ( E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> ( E. s ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
19	4, 5, 18	3bitr4ri 212	. . 3 |- ( E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> t e. ( ( `' F " B ) i^i A ) )
20	2, 19	bitri 183	. 2 |- ( t e. ( `' ( F |` A ) " B ) <-> t e. ( ( `' F " B ) i^i A ) )
21	20	eqriv 2162	1 |- ( `' ( F |` A ) " B ) = ( ( `' F " B ) i^i A )

Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   i^i cin 3115   <.cop 3579   `'ccnv 4603   |` cres 4606   "cima 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by:  cnrest  12875