Theorem cnvresima 5120

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 2742	. . . 4 |- t e. _V
2	1	elima3 4979	. . 3 |- ( t e. ( `' ( F |` A ) " B ) <-> E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) )
3	1	elima3 4979	. . . . 5 |- ( t e. ( `' F " B ) <-> E. s ( s e. B /\ <. s , t >. e. `' F ) )
4	3	anbi1i 458	. . . 4 |- ( ( t e. ( `' F " B ) /\ t e. A ) <-> ( E. s ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
5		elin 3320	. . . 4 |- ( t e. ( ( `' F " B ) i^i A ) <-> ( t e. ( `' F " B ) /\ t e. A ) )
6		vex 2742	. . . . . . . . . 10 |- s e. _V
7	6, 1	opelcnv 4811	. . . . . . . . 9 |- ( <. s , t >. e. `' ( F |` A ) <-> <. t , s >. e. ( F |` A ) )
8	6	opelres 4914	. . . . . . . . . 10 |- ( <. t , s >. e. ( F |` A ) <-> ( <. t , s >. e. F /\ t e. A ) )
9	6, 1	opelcnv 4811	. . . . . . . . . . 11 |- ( <. s , t >. e. `' F <-> <. t , s >. e. F )
10	9	anbi1i 458	. . . . . . . . . 10 |- ( ( <. s , t >. e. `' F /\ t e. A ) <-> ( <. t , s >. e. F /\ t e. A ) )
11	8, 10	bitr4i 187	. . . . . . . . 9 |- ( <. t , s >. e. ( F |` A ) <-> ( <. s , t >. e. `' F /\ t e. A ) )
12	7, 11	bitri 184	. . . . . . . 8 |- ( <. s , t >. e. `' ( F |` A ) <-> ( <. s , t >. e. `' F /\ t e. A ) )
13	12	anbi2i 457	. . . . . . 7 |- ( ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> ( s e. B /\ ( <. s , t >. e. `' F /\ t e. A ) ) )
14		anass 401	. . . . . . 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 187	. . . . . 6 |- ( ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> ( ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
16	15	exbii 1605	. . . . 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 1902	. . . . 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 184	. . . 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 213	. . 3 |- ( E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> t e. ( ( `' F " B ) i^i A ) )
20	2, 19	bitri 184	. 2 |- ( t e. ( `' ( F |` A ) " B ) <-> t e. ( ( `' F " B ) i^i A ) )
21	20	eqriv 2174	1 |- ( `' ( F |` A ) " B ) = ( ( `' F " B ) i^i A )

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1492   e. wcel 2148   i^i cin 3130   <.cop 3597   `'ccnv 4627   |` cres 4630   "cima 4631

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641

This theorem is referenced by:  cnrest  13820