Theorem brdomaing 32042

Description: Closed form of brdomain 32040. (Contributed by Scott Fenton, 2-May-2014.)

Assertion

Ref	Expression
brdomaing	|- ( ( A e. V /\ B e. W ) -> ( ADomain B <-> B = dom A ) )

Proof of Theorem brdomaing

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4656	. . 3 |- ( a = A -> ( aDomain b <-> ADomain b ) )
2		dmeq 5324	. . . 4 |- ( a = A -> dom a = dom A )
3	2	eqeq2d 2632	. . 3 |- ( a = A -> ( b = dom a <-> b = dom A ) )
4	1, 3	bibi12d 335	. 2 |- ( a = A -> ( ( aDomain b <-> b = dom a ) <-> ( ADomain b <-> b = dom A ) ) )
5		breq2 4657	. . 3 |- ( b = B -> ( ADomain b <-> ADomain B ) )
6		eqeq1 2626	. . 3 |- ( b = B -> ( b = dom A <-> B = dom A ) )
7	5, 6	bibi12d 335	. 2 |- ( b = B -> ( ( ADomain b <-> b = dom A ) <-> ( ADomain B <-> B = dom A ) ) )
8		vex 3203	. . 3 |- a e. _V
9		vex 3203	. . 3 |- b e. _V
10	8, 9	brdomain 32040	. 2 |- ( aDomain b <-> b = dom a )
11	4, 7, 10	vtocl2g 3270	1 |- ( ( A e. V /\ B e. W ) -> ( ADomain B <-> B = dom A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   dom cdm 5114  Domaincdomain 31950

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-image 31971  df-domain 31974

This theorem is referenced by: (None)