Theorem dfres3 5403

Description: Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
dfres3	|- ( A |` B ) = ( A i^i ( B X. ran A ) )

Proof of Theorem dfres3

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-res 5126	. 2 |- ( A |` B ) = ( A i^i ( B X. _V ) )
2		eleq1 2689	. . . . . . . . . 10 |- ( x = <. y , z >. -> ( x e. A <-> <. y , z >. e. A ) )
3		vex 3203	. . . . . . . . . . . 12 |- z e. _V
4	3	biantru 526	. . . . . . . . . . 11 |- ( y e. B <-> ( y e. B /\ z e. _V ) )
5		vex 3203	. . . . . . . . . . . . 13 |- y e. _V
6	5, 3	opelrn 5357	. . . . . . . . . . . 12 |- ( <. y , z >. e. A -> z e. ran A )
7	6	biantrud 528	. . . . . . . . . . 11 |- ( <. y , z >. e. A -> ( y e. B <-> ( y e. B /\ z e. ran A ) ) )
8	4, 7	syl5bbr 274	. . . . . . . . . 10 |- ( <. y , z >. e. A -> ( ( y e. B /\ z e. _V ) <-> ( y e. B /\ z e. ran A ) ) )
9	2, 8	syl6bi 243	. . . . . . . . 9 |- ( x = <. y , z >. -> ( x e. A -> ( ( y e. B /\ z e. _V ) <-> ( y e. B /\ z e. ran A ) ) ) )
10	9	com12 32	. . . . . . . 8 |- ( x e. A -> ( x = <. y , z >. -> ( ( y e. B /\ z e. _V ) <-> ( y e. B /\ z e. ran A ) ) ) )
11	10	pm5.32d 671	. . . . . . 7 |- ( x e. A -> ( ( x = <. y , z >. /\ ( y e. B /\ z e. _V ) ) <-> ( x = <. y , z >. /\ ( y e. B /\ z e. ran A ) ) ) )
12	11	2exbidv 1852	. . . . . 6 |- ( x e. A -> ( E. y E. z ( x = <. y , z >. /\ ( y e. B /\ z e. _V ) ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. B /\ z e. ran A ) ) ) )
13		elxp 5131	. . . . . 6 |- ( x e. ( B X. _V ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. B /\ z e. _V ) ) )
14		elxp 5131	. . . . . 6 |- ( x e. ( B X. ran A ) <-> E. y E. z ( x = <. y , z >. /\ ( y e. B /\ z e. ran A ) ) )
15	12, 13, 14	3bitr4g 303	. . . . 5 |- ( x e. A -> ( x e. ( B X. _V ) <-> x e. ( B X. ran A ) ) )
16	15	pm5.32i 669	. . . 4 |- ( ( x e. A /\ x e. ( B X. _V ) ) <-> ( x e. A /\ x e. ( B X. ran A ) ) )
17		elin 3796	. . . 4 |- ( x e. ( A i^i ( B X. ran A ) ) <-> ( x e. A /\ x e. ( B X. ran A ) ) )
18	16, 17	bitr4i 267	. . 3 |- ( ( x e. A /\ x e. ( B X. _V ) ) <-> x e. ( A i^i ( B X. ran A ) ) )
19	18	ineqri 3806	. 2 |- ( A i^i ( B X. _V ) ) = ( A i^i ( B X. ran A ) )
20	1, 19	eqtri 2644	1 |- ( A |` B ) = ( A i^i ( B X. ran A ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   i^i cin 3573   <.cop 4183   X. cxp 5112   ran crn 5115   |` cres 5116

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-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

This theorem is referenced by:  brrestrict  32056  dfrel6  34115