Theorem cnvresima 4810

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 2560	. . . 4 |- t e. _V
2	1	elima3 4675	. . 3 |- ( t e. ( `' ( F |` A ) " B ) <-> E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) )
3	1	elima3 4675	. . . . 5 |- ( t e. ( `' F " B ) <-> E. s ( s e. B /\ <. s , t >. e. `' F ) )
4	3	anbi1i 431	. . . 4 |- ( ( t e. ( `' F " B ) /\ t e. A ) <-> ( E. s ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
5		elin 3126	. . . 4 |- ( t e. ( ( `' F " B ) i^i A ) <-> ( t e. ( `' F " B ) /\ t e. A ) )
6		vex 2560	. . . . . . . . . 10 |- s e. _V
7	6, 1	opelcnv 4517	. . . . . . . . 9 |- ( <. s , t >. e. `' ( F |` A ) <-> <. t , s >. e. ( F |` A ) )
8	6	opelres 4617	. . . . . . . . . 10 |- ( <. t , s >. e. ( F |` A ) <-> ( <. t , s >. e. F /\ t e. A ) )
9	6, 1	opelcnv 4517	. . . . . . . . . . 11 |- ( <. s , t >. e. `' F <-> <. t , s >. e. F )
10	9	anbi1i 431	. . . . . . . . . 10 |- ( ( <. s , t >. e. `' F /\ t e. A ) <-> ( <. t , s >. e. F /\ t e. A ) )
11	8, 10	bitr4i 176	. . . . . . . . 9 |- ( <. t , s >. e. ( F |` A ) <-> ( <. s , t >. e. `' F /\ t e. A ) )
12	7, 11	bitri 173	. . . . . . . 8 |- ( <. s , t >. e. `' ( F |` A ) <-> ( <. s , t >. e. `' F /\ t e. A ) )
13	12	anbi2i 430	. . . . . . 7 |- ( ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> ( s e. B /\ ( <. s , t >. e. `' F /\ t e. A ) ) )
14		anass 381	. . . . . . 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 176	. . . . . 6 |- ( ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> ( ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
16	15	exbii 1496	. . . . 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 1782	. . . . 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 173	. . . 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 202	. . 3 |- ( E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> t e. ( ( `' F " B ) i^i A ) )
20	2, 19	bitri 173	. 2 |- ( t e. ( `' ( F |` A ) " B ) <-> t e. ( ( `' F " B ) i^i A ) )
21	20	eqriv 2037	1 |- ( `' ( F |` A ) " B ) = ( ( `' F " B ) i^i A )

Syntax hints:   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   i^i cin 2916   <.cop 3378   `'ccnv 4344   |` cres 4347   "cima 4348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358

This theorem is referenced by: (None)