Theorem cnvresima 5224

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 2803	. . . 4 |- t e. _V
2	1	elima3 5081	. . 3 |- ( t e. ( `' ( F |` A ) " B ) <-> E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) )
3	1	elima3 5081	. . . . 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 3388	. . . 4 |- ( t e. ( ( `' F " B ) i^i A ) <-> ( t e. ( `' F " B ) /\ t e. A ) )
6		vex 2803	. . . . . . . . . 10 |- s e. _V
7	6, 1	opelcnv 4910	. . . . . . . . 9 |- ( <. s , t >. e. `' ( F |` A ) <-> <. t , s >. e. ( F |` A ) )
8	6	opelres 5016	. . . . . . . . . 10 |- ( <. t , s >. e. ( F |` A ) <-> ( <. t , s >. e. F /\ t e. A ) )
9	6, 1	opelcnv 4910	. . . . . . . . . . 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 1651	. . . . 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 1949	. . . . 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 2226	1 |- ( `' ( F |` A ) " B ) = ( ( `' F " B ) i^i A )

Syntax hints:   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   i^i cin 3197   <.cop 3670   `'ccnv 4722   |` cres 4725   "cima 4726

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736

This theorem is referenced by:  cnrest  14949