Theorem relop 5272

Description: A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
relop.1	|- A e. _V
relop.2	|- B e. _V

Assertion

Ref	Expression
relop	|- ( Rel <. A , B >. <-> E. x E. y ( A = { x } /\ B = { x , y } ) )

Distinct variable groups:   x, y, A   x, B, y

Proof of Theorem relop

Dummy variables w v z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rel 5121	. 2 |- ( Rel <. A , B >. <-> <. A , B >. C_ ( _V X. _V ) )
2		dfss2 3591	. . . . 5 |- ( <. A , B >. C_ ( _V X. _V ) <-> A. z ( z e. <. A , B >. -> z e. ( _V X. _V ) ) )
3		relop.1	. . . . . . . . . 10 |- A e. _V
4		relop.2	. . . . . . . . . 10 |- B e. _V
5	3, 4	elop 4935	. . . . . . . . 9 |- ( z e. <. A , B >. <-> ( z = { A } \/ z = { A , B } ) )
6		elvv 5177	. . . . . . . . 9 |- ( z e. ( _V X. _V ) <-> E. x E. y z = <. x , y >. )
7	5, 6	imbi12i 340	. . . . . . . 8 |- ( ( z e. <. A , B >. -> z e. ( _V X. _V ) ) <-> ( ( z = { A } \/ z = { A , B } ) -> E. x E. y z = <. x , y >. ) )
8		jaob 822	. . . . . . . 8 |- ( ( ( z = { A } \/ z = { A , B } ) -> E. x E. y z = <. x , y >. ) <-> ( ( z = { A } -> E. x E. y z = <. x , y >. ) /\ ( z = { A , B } -> E. x E. y z = <. x , y >. ) ) )
9	7, 8	bitri 264	. . . . . . 7 |- ( ( z e. <. A , B >. -> z e. ( _V X. _V ) ) <-> ( ( z = { A } -> E. x E. y z = <. x , y >. ) /\ ( z = { A , B } -> E. x E. y z = <. x , y >. ) ) )
10	9	albii 1747	. . . . . 6 |- ( A. z ( z e. <. A , B >. -> z e. ( _V X. _V ) ) <-> A. z ( ( z = { A } -> E. x E. y z = <. x , y >. ) /\ ( z = { A , B } -> E. x E. y z = <. x , y >. ) ) )
11		19.26 1798	. . . . . 6 |- ( A. z ( ( z = { A } -> E. x E. y z = <. x , y >. ) /\ ( z = { A , B } -> E. x E. y z = <. x , y >. ) ) <-> ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) /\ A. z ( z = { A , B } -> E. x E. y z = <. x , y >. ) ) )
12	10, 11	bitri 264	. . . . 5 |- ( A. z ( z e. <. A , B >. -> z e. ( _V X. _V ) ) <-> ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) /\ A. z ( z = { A , B } -> E. x E. y z = <. x , y >. ) ) )
13	2, 12	bitri 264	. . . 4 |- ( <. A , B >. C_ ( _V X. _V ) <-> ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) /\ A. z ( z = { A , B } -> E. x E. y z = <. x , y >. ) ) )
14		snex 4908	. . . . . . 7 |- { A } e. _V
15		eqeq1 2626	. . . . . . . 8 |- ( z = { A } -> ( z = { A } <-> { A } = { A } ) )
16		eqeq1 2626	. . . . . . . . . 10 |- ( z = { A } -> ( z = <. x , y >. <-> { A } = <. x , y >. ) )
17		eqcom 2629	. . . . . . . . . . 11 |- ( { A } = <. x , y >. <-> <. x , y >. = { A } )
18		vex 3203	. . . . . . . . . . . 12 |- x e. _V
19		vex 3203	. . . . . . . . . . . 12 |- y e. _V
20	18, 19, 3	opeqsn 4967	. . . . . . . . . . 11 |- ( <. x , y >. = { A } <-> ( x = y /\ A = { x } ) )
21	17, 20	bitri 264	. . . . . . . . . 10 |- ( { A } = <. x , y >. <-> ( x = y /\ A = { x } ) )
22	16, 21	syl6bb 276	. . . . . . . . 9 |- ( z = { A } -> ( z = <. x , y >. <-> ( x = y /\ A = { x } ) ) )
23	22	2exbidv 1852	. . . . . . . 8 |- ( z = { A } -> ( E. x E. y z = <. x , y >. <-> E. x E. y ( x = y /\ A = { x } ) ) )
24	15, 23	imbi12d 334	. . . . . . 7 |- ( z = { A } -> ( ( z = { A } -> E. x E. y z = <. x , y >. ) <-> ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) ) )
25	14, 24	spcv 3299	. . . . . 6 |- ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) -> ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) )
26		sneq 4187	. . . . . . . . 9 |- ( w = x -> { w } = { x } )
27	26	eqeq2d 2632	. . . . . . . 8 |- ( w = x -> ( A = { w } <-> A = { x } ) )
28	27	cbvexv 2275	. . . . . . 7 |- ( E. w A = { w } <-> E. x A = { x } )
29		ax6evr 1942	. . . . . . . . 9 |- E. y x = y
30		19.41v 1914	. . . . . . . . 9 |- ( E. y ( x = y /\ A = { x } ) <-> ( E. y x = y /\ A = { x } ) )
31	29, 30	mpbiran 953	. . . . . . . 8 |- ( E. y ( x = y /\ A = { x } ) <-> A = { x } )
32	31	exbii 1774	. . . . . . 7 |- ( E. x E. y ( x = y /\ A = { x } ) <-> E. x A = { x } )
33		eqid 2622	. . . . . . . 8 |- { A } = { A }
34	33	a1bi 352	. . . . . . 7 |- ( E. x E. y ( x = y /\ A = { x } ) <-> ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) )
35	28, 32, 34	3bitr2ri 289	. . . . . 6 |- ( ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) <-> E. w A = { w } )
36	25, 35	sylib 208	. . . . 5 |- ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) -> E. w A = { w } )
37		eqid 2622	. . . . . 6 |- { A , B } = { A , B }
38		prex 4909	. . . . . . 7 |- { A , B } e. _V
39		eqeq1 2626	. . . . . . . 8 |- ( z = { A , B } -> ( z = { A , B } <-> { A , B } = { A , B } ) )
40		eqeq1 2626	. . . . . . . . 9 |- ( z = { A , B } -> ( z = <. x , y >. <-> { A , B } = <. x , y >. ) )
41	40	2exbidv 1852	. . . . . . . 8 |- ( z = { A , B } -> ( E. x E. y z = <. x , y >. <-> E. x E. y { A , B } = <. x , y >. ) )
42	39, 41	imbi12d 334	. . . . . . 7 |- ( z = { A , B } -> ( ( z = { A , B } -> E. x E. y z = <. x , y >. ) <-> ( { A , B } = { A , B } -> E. x E. y { A , B } = <. x , y >. ) ) )
43	38, 42	spcv 3299	. . . . . 6 |- ( A. z ( z = { A , B } -> E. x E. y z = <. x , y >. ) -> ( { A , B } = { A , B } -> E. x E. y { A , B } = <. x , y >. ) )
44	37, 43	mpi 20	. . . . 5 |- ( A. z ( z = { A , B } -> E. x E. y z = <. x , y >. ) -> E. x E. y { A , B } = <. x , y >. )
45		eqcom 2629	. . . . . . . . . 10 |- ( { A , B } = <. x , y >. <-> <. x , y >. = { A , B } )
46	18, 19, 3, 4	opeqpr 4968	. . . . . . . . . 10 |- ( <. x , y >. = { A , B } <-> ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) )
47	45, 46	bitri 264	. . . . . . . . 9 |- ( { A , B } = <. x , y >. <-> ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) )
48		idd 24	. . . . . . . . . 10 |- ( A = { w } -> ( ( A = { x } /\ B = { x , y } ) -> ( A = { x } /\ B = { x , y } ) ) )
49		eqtr2 2642	. . . . . . . . . . . . . 14 |- ( ( A = { x , y } /\ A = { w } ) -> { x , y } = { w } )
50		vex 3203	. . . . . . . . . . . . . . . 16 |- w e. _V
51	18, 19, 50	preqsn 4393	. . . . . . . . . . . . . . 15 |- ( { x , y } = { w } <-> ( x = y /\ y = w ) )
52	51	simplbi 476	. . . . . . . . . . . . . 14 |- ( { x , y } = { w } -> x = y )
53	49, 52	syl 17	. . . . . . . . . . . . 13 |- ( ( A = { x , y } /\ A = { w } ) -> x = y )
54		dfsn2 4190	. . . . . . . . . . . . . . . . . . . 20 |- { x } = { x , x }
55		preq2 4269	. . . . . . . . . . . . . . . . . . . 20 |- ( x = y -> { x , x } = { x , y } )
56	54, 55	syl5req 2669	. . . . . . . . . . . . . . . . . . 19 |- ( x = y -> { x , y } = { x } )
57	56	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( A = { x , y } <-> A = { x } ) )
58	54, 55	syl5eq 2668	. . . . . . . . . . . . . . . . . . 19 |- ( x = y -> { x } = { x , y } )
59	58	eqeq2d 2632	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( B = { x } <-> B = { x , y } ) )
60	57, 59	anbi12d 747	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( ( A = { x , y } /\ B = { x } ) <-> ( A = { x } /\ B = { x , y } ) ) )
61	60	biimpd 219	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( ( A = { x , y } /\ B = { x } ) -> ( A = { x } /\ B = { x , y } ) ) )
62	61	expd 452	. . . . . . . . . . . . . . 15 |- ( x = y -> ( A = { x , y } -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
63	62	com12 32	. . . . . . . . . . . . . 14 |- ( A = { x , y } -> ( x = y -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
64	63	adantr 481	. . . . . . . . . . . . 13 |- ( ( A = { x , y } /\ A = { w } ) -> ( x = y -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
65	53, 64	mpd 15	. . . . . . . . . . . 12 |- ( ( A = { x , y } /\ A = { w } ) -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) )
66	65	expcom 451	. . . . . . . . . . 11 |- ( A = { w } -> ( A = { x , y } -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
67	66	impd 447	. . . . . . . . . 10 |- ( A = { w } -> ( ( A = { x , y } /\ B = { x } ) -> ( A = { x } /\ B = { x , y } ) ) )
68	48, 67	jaod 395	. . . . . . . . 9 |- ( A = { w } -> ( ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) -> ( A = { x } /\ B = { x , y } ) ) )
69	47, 68	syl5bi 232	. . . . . . . 8 |- ( A = { w } -> ( { A , B } = <. x , y >. -> ( A = { x } /\ B = { x , y } ) ) )
70	69	2eximdv 1848	. . . . . . 7 |- ( A = { w } -> ( E. x E. y { A , B } = <. x , y >. -> E. x E. y ( A = { x } /\ B = { x , y } ) ) )
71	70	exlimiv 1858	. . . . . 6 |- ( E. w A = { w } -> ( E. x E. y { A , B } = <. x , y >. -> E. x E. y ( A = { x } /\ B = { x , y } ) ) )
72	71	imp 445	. . . . 5 |- ( ( E. w A = { w } /\ E. x E. y { A , B } = <. x , y >. ) -> E. x E. y ( A = { x } /\ B = { x , y } ) )
73	36, 44, 72	syl2an 494	. . . 4 |- ( ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) /\ A. z ( z = { A , B } -> E. x E. y z = <. x , y >. ) ) -> E. x E. y ( A = { x } /\ B = { x , y } ) )
74	13, 73	sylbi 207	. . 3 |- ( <. A , B >. C_ ( _V X. _V ) -> E. x E. y ( A = { x } /\ B = { x , y } ) )
75		simpr 477	. . . . . . . . . . 11 |- ( ( A = { x } /\ z = { A } ) -> z = { A } )
76		equid 1939	. . . . . . . . . . . . . 14 |- x = x
77	76	jctl 564	. . . . . . . . . . . . 13 |- ( A = { x } -> ( x = x /\ A = { x } ) )
78	18, 18, 3	opeqsn 4967	. . . . . . . . . . . . 13 |- ( <. x , x >. = { A } <-> ( x = x /\ A = { x } ) )
79	77, 78	sylibr 224	. . . . . . . . . . . 12 |- ( A = { x } -> <. x , x >. = { A } )
80	79	adantr 481	. . . . . . . . . . 11 |- ( ( A = { x } /\ z = { A } ) -> <. x , x >. = { A } )
81	75, 80	eqtr4d 2659	. . . . . . . . . 10 |- ( ( A = { x } /\ z = { A } ) -> z = <. x , x >. )
82		opeq12 4404	. . . . . . . . . . . 12 |- ( ( w = x /\ v = x ) -> <. w , v >. = <. x , x >. )
83	82	eqeq2d 2632	. . . . . . . . . . 11 |- ( ( w = x /\ v = x ) -> ( z = <. w , v >. <-> z = <. x , x >. ) )
84	18, 18, 83	spc2ev 3301	. . . . . . . . . 10 |- ( z = <. x , x >. -> E. w E. v z = <. w , v >. )
85	81, 84	syl 17	. . . . . . . . 9 |- ( ( A = { x } /\ z = { A } ) -> E. w E. v z = <. w , v >. )
86	85	adantlr 751	. . . . . . . 8 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A } ) -> E. w E. v z = <. w , v >. )
87		preq12 4270	. . . . . . . . . . . 12 |- ( ( A = { x } /\ B = { x , y } ) -> { A , B } = { { x } , { x , y } } )
88	87	eqeq2d 2632	. . . . . . . . . . 11 |- ( ( A = { x } /\ B = { x , y } ) -> ( z = { A , B } <-> z = { { x } , { x , y } } ) )
89	88	biimpa 501	. . . . . . . . . 10 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> z = { { x } , { x , y } } )
90	18, 19	dfop 4401	. . . . . . . . . 10 |- <. x , y >. = { { x } , { x , y } }
91	89, 90	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> z = <. x , y >. )
92		opeq12 4404	. . . . . . . . . . 11 |- ( ( w = x /\ v = y ) -> <. w , v >. = <. x , y >. )
93	92	eqeq2d 2632	. . . . . . . . . 10 |- ( ( w = x /\ v = y ) -> ( z = <. w , v >. <-> z = <. x , y >. ) )
94	18, 19, 93	spc2ev 3301	. . . . . . . . 9 |- ( z = <. x , y >. -> E. w E. v z = <. w , v >. )
95	91, 94	syl 17	. . . . . . . 8 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> E. w E. v z = <. w , v >. )
96	86, 95	jaodan 826	. . . . . . 7 |- ( ( ( A = { x } /\ B = { x , y } ) /\ ( z = { A } \/ z = { A , B } ) ) -> E. w E. v z = <. w , v >. )
97	96	ex 450	. . . . . 6 |- ( ( A = { x } /\ B = { x , y } ) -> ( ( z = { A } \/ z = { A , B } ) -> E. w E. v z = <. w , v >. ) )
98		elvv 5177	. . . . . 6 |- ( z e. ( _V X. _V ) <-> E. w E. v z = <. w , v >. )
99	97, 5, 98	3imtr4g 285	. . . . 5 |- ( ( A = { x } /\ B = { x , y } ) -> ( z e. <. A , B >. -> z e. ( _V X. _V ) ) )
100	99	ssrdv 3609	. . . 4 |- ( ( A = { x } /\ B = { x , y } ) -> <. A , B >. C_ ( _V X. _V ) )
101	100	exlimivv 1860	. . 3 |- ( E. x E. y ( A = { x } /\ B = { x , y } ) -> <. A , B >. C_ ( _V X. _V ) )
102	74, 101	impbii 199	. 2 |- ( <. A , B >. C_ ( _V X. _V ) <-> E. x E. y ( A = { x } /\ B = { x , y } ) )
103	1, 102	bitri 264	1 |- ( Rel <. A , B >. <-> E. x E. y ( A = { x } /\ B = { x , y } ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   <.cop 4183   X. cxp 5112   Rel wrel 5119

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-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-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  funopg  5922