Theorem brcup 32046

Description: Binary relation form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brcup.1	|- A e. _V
brcup.2	|- B e. _V
brcup.3	|- C e. _V

Assertion

Ref	Expression
brcup	|- ( <. A , B >.Cup C <-> C = ( A u. B ) )

Proof of Theorem brcup

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

Step	Hyp	Ref	Expression
1		opex 4932	. 2 |- <. A , B >. e. _V
2		brcup.3	. 2 |- C e. _V
3		df-cup 31976	. 2 |- Cup = ( ( ( _V X. _V ) X. _V ) \ ran ( ( _V (x) _E ) /_\ ( ( ( `' 1st o. _E ) u. ( `' 2nd o. _E ) ) (x) _V ) ) )
4		brcup.1	. . . 4 |- A e. _V
5		brcup.2	. . . 4 |- B e. _V
6	4, 5	opelvv 5166	. . 3 |- <. A , B >. e. ( _V X. _V )
7		brxp 5147	. . 3 |- ( <. A , B >. ( ( _V X. _V ) X. _V ) C <-> ( <. A , B >. e. ( _V X. _V ) /\ C e. _V ) )
8	6, 2, 7	mpbir2an 955	. 2 |- <. A , B >. ( ( _V X. _V ) X. _V ) C
9		epel 5032	. . . . . . 7 |- ( x _E y <-> x e. y )
10		vex 3203	. . . . . . . . 9 |- y e. _V
11	10, 1	brcnv 5305	. . . . . . . 8 |- ( y `' 1st <. A , B >. <-> <. A , B >. 1st y )
12	4, 5	br1steq 31670	. . . . . . . 8 |- ( <. A , B >. 1st y <-> y = A )
13	11, 12	bitri 264	. . . . . . 7 |- ( y `' 1st <. A , B >. <-> y = A )
14	9, 13	anbi12ci 734	. . . . . 6 |- ( ( x _E y /\ y `' 1st <. A , B >. ) <-> ( y = A /\ x e. y ) )
15	14	exbii 1774	. . . . 5 |- ( E. y ( x _E y /\ y `' 1st <. A , B >. ) <-> E. y ( y = A /\ x e. y ) )
16		vex 3203	. . . . . 6 |- x e. _V
17	16, 1	brco 5292	. . . . 5 |- ( x ( `' 1st o. _E ) <. A , B >. <-> E. y ( x _E y /\ y `' 1st <. A , B >. ) )
18	4	clel3 3341	. . . . 5 |- ( x e. A <-> E. y ( y = A /\ x e. y ) )
19	15, 17, 18	3bitr4i 292	. . . 4 |- ( x ( `' 1st o. _E ) <. A , B >. <-> x e. A )
20	10, 1	brcnv 5305	. . . . . . . 8 |- ( y `' 2nd <. A , B >. <-> <. A , B >. 2nd y )
21	4, 5	br2ndeq 31671	. . . . . . . 8 |- ( <. A , B >. 2nd y <-> y = B )
22	20, 21	bitri 264	. . . . . . 7 |- ( y `' 2nd <. A , B >. <-> y = B )
23	9, 22	anbi12ci 734	. . . . . 6 |- ( ( x _E y /\ y `' 2nd <. A , B >. ) <-> ( y = B /\ x e. y ) )
24	23	exbii 1774	. . . . 5 |- ( E. y ( x _E y /\ y `' 2nd <. A , B >. ) <-> E. y ( y = B /\ x e. y ) )
25	16, 1	brco 5292	. . . . 5 |- ( x ( `' 2nd o. _E ) <. A , B >. <-> E. y ( x _E y /\ y `' 2nd <. A , B >. ) )
26	5	clel3 3341	. . . . 5 |- ( x e. B <-> E. y ( y = B /\ x e. y ) )
27	24, 25, 26	3bitr4i 292	. . . 4 |- ( x ( `' 2nd o. _E ) <. A , B >. <-> x e. B )
28	19, 27	orbi12i 543	. . 3 |- ( ( x ( `' 1st o. _E ) <. A , B >. \/ x ( `' 2nd o. _E ) <. A , B >. ) <-> ( x e. A \/ x e. B ) )
29		brun 4703	. . 3 |- ( x ( ( `' 1st o. _E ) u. ( `' 2nd o. _E ) ) <. A , B >. <-> ( x ( `' 1st o. _E ) <. A , B >. \/ x ( `' 2nd o. _E ) <. A , B >. ) )
30		elun 3753	. . 3 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
31	28, 29, 30	3bitr4ri 293	. 2 |- ( x e. ( A u. B ) <-> x ( ( `' 1st o. _E ) u. ( `' 2nd o. _E ) ) <. A , B >. )
32	1, 2, 3, 8, 31	brtxpsd3 32003	1 |- ( <. A , B >.Cup C <-> C = ( A u. B ) )

Syntax hints:   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   u. cun 3572   <.cop 4183   class class class wbr 4653   _E cep 5028   X. cxp 5112   `'ccnv 5113   o. ccom 5118   1stc1st 7166   2ndc2nd 7167  Cupccup 31953

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-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-cup 31976

This theorem is referenced by:  brsuccf  32048