Theorem cnvresima 5623

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		19.41v 1914	. . . 4 |- ( E. s ( ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) <-> ( E. s ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
2		vex 3203	. . . . . . . 8 |- s e. _V
3	2	opelres 5401	. . . . . . 7 |- ( <. t , s >. e. ( F |` A ) <-> ( <. t , s >. e. F /\ t e. A ) )
4		vex 3203	. . . . . . . 8 |- t e. _V
5	2, 4	opelcnv 5304	. . . . . . 7 |- ( <. s , t >. e. `' ( F |` A ) <-> <. t , s >. e. ( F |` A ) )
6	2, 4	opelcnv 5304	. . . . . . . 8 |- ( <. s , t >. e. `' F <-> <. t , s >. e. F )
7	6	anbi1i 731	. . . . . . 7 |- ( ( <. s , t >. e. `' F /\ t e. A ) <-> ( <. t , s >. e. F /\ t e. A ) )
8	3, 5, 7	3bitr4i 292	. . . . . 6 |- ( <. s , t >. e. `' ( F |` A ) <-> ( <. s , t >. e. `' F /\ t e. A ) )
9	8	bianass 842	. . . . 5 |- ( ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> ( ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
10	9	exbii 1774	. . . 4 |- ( E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> E. s ( ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
11	4	elima3 5473	. . . . 5 |- ( t e. ( `' F " B ) <-> E. s ( s e. B /\ <. s , t >. e. `' F ) )
12	11	anbi1i 731	. . . 4 |- ( ( t e. ( `' F " B ) /\ t e. A ) <-> ( E. s ( s e. B /\ <. s , t >. e. `' F ) /\ t e. A ) )
13	1, 10, 12	3bitr4i 292	. . 3 |- ( E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) <-> ( t e. ( `' F " B ) /\ t e. A ) )
14	4	elima3 5473	. . 3 |- ( t e. ( `' ( F |` A ) " B ) <-> E. s ( s e. B /\ <. s , t >. e. `' ( F |` A ) ) )
15		elin 3796	. . 3 |- ( t e. ( ( `' F " B ) i^i A ) <-> ( t e. ( `' F " B ) /\ t e. A ) )
16	13, 14, 15	3bitr4i 292	. 2 |- ( t e. ( `' ( F |` A ) " B ) <-> t e. ( ( `' F " B ) i^i A ) )
17	16	eqriv 2619	1 |- ( `' ( F |` A ) " B ) = ( ( `' F " B ) i^i A )

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   i^i cin 3573   <.cop 4183   `'ccnv 5113   |` cres 5116   "cima 5117

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  fimacnvinrn  6348  ramub2  15718  ramub1lem2  15731  cnrest  21089  kgencn  21359  kgencn3  21361  xkoptsub  21457  qtopres  21501  qtoprest  21520  mbfid  23403  mbfres  23411  1stpreima  29484  2ndpreima  29485  cvmsss2  31256  lmhmlnmsplit  37657