Theorem relop 4816

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 4670	. 2 |- ( Rel <. A , B >. <-> <. A , B >. C_ ( _V X. _V ) )
2		dfss2 3172	. . . . 5 |- ( <. A , B >. C_ ( _V X. _V ) <-> A. z ( z e. <. A , B >. -> z e. ( _V X. _V ) ) )
3		vex 2766	. . . . . . . . . 10 |- z e. _V
4		relop.1	. . . . . . . . . 10 |- A e. _V
5		relop.2	. . . . . . . . . 10 |- B e. _V
6	3, 4, 5	elop 4264	. . . . . . . . 9 |- ( z e. <. A , B >. <-> ( z = { A } \/ z = { A , B } ) )
7		elvv 4725	. . . . . . . . 9 |- ( z e. ( _V X. _V ) <-> E. x E. y z = <. x , y >. )
8	6, 7	imbi12i 239	. . . . . . . 8 |- ( ( z e. <. A , B >. -> z e. ( _V X. _V ) ) <-> ( ( z = { A } \/ z = { A , B } ) -> E. x E. y z = <. x , y >. ) )
9		jaob 711	. . . . . . . 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 >. ) ) )
10	8, 9	bitri 184	. . . . . . 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 >. ) ) )
11	10	albii 1484	. . . . . 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 >. ) ) )
12		19.26 1495	. . . . . 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 >. ) ) )
13	11, 12	bitri 184	. . . . 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 >. ) ) )
14	2, 13	bitri 184	. . . 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 >. ) ) )
15	4	snex 4218	. . . . . . 7 |- { A } e. _V
16		eqeq1 2203	. . . . . . . 8 |- ( z = { A } -> ( z = { A } <-> { A } = { A } ) )
17		eqeq1 2203	. . . . . . . . . 10 |- ( z = { A } -> ( z = <. x , y >. <-> { A } = <. x , y >. ) )
18		eqcom 2198	. . . . . . . . . . 11 |- ( { A } = <. x , y >. <-> <. x , y >. = { A } )
19		vex 2766	. . . . . . . . . . . 12 |- x e. _V
20		vex 2766	. . . . . . . . . . . 12 |- y e. _V
21	19, 20, 4	opeqsn 4285	. . . . . . . . . . 11 |- ( <. x , y >. = { A } <-> ( x = y /\ A = { x } ) )
22	18, 21	bitri 184	. . . . . . . . . 10 |- ( { A } = <. x , y >. <-> ( x = y /\ A = { x } ) )
23	17, 22	bitrdi 196	. . . . . . . . 9 |- ( z = { A } -> ( z = <. x , y >. <-> ( x = y /\ A = { x } ) ) )
24	23	2exbidv 1882	. . . . . . . 8 |- ( z = { A } -> ( E. x E. y z = <. x , y >. <-> E. x E. y ( x = y /\ A = { x } ) ) )
25	16, 24	imbi12d 234	. . . . . . 7 |- ( z = { A } -> ( ( z = { A } -> E. x E. y z = <. x , y >. ) <-> ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) ) )
26	15, 25	spcv 2858	. . . . . 6 |- ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) -> ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) )
27		sneq 3633	. . . . . . . . 9 |- ( w = x -> { w } = { x } )
28	27	eqeq2d 2208	. . . . . . . 8 |- ( w = x -> ( A = { w } <-> A = { x } ) )
29	28	cbvexv 1933	. . . . . . 7 |- ( E. w A = { w } <-> E. x A = { x } )
30		a9ev 1711	. . . . . . . . . 10 |- E. y y = x
31		equcom 1720	. . . . . . . . . . 11 |- ( y = x <-> x = y )
32	31	exbii 1619	. . . . . . . . . 10 |- ( E. y y = x <-> E. y x = y )
33	30, 32	mpbi 145	. . . . . . . . 9 |- E. y x = y
34		19.41v 1917	. . . . . . . . 9 |- ( E. y ( x = y /\ A = { x } ) <-> ( E. y x = y /\ A = { x } ) )
35	33, 34	mpbiran 942	. . . . . . . 8 |- ( E. y ( x = y /\ A = { x } ) <-> A = { x } )
36	35	exbii 1619	. . . . . . 7 |- ( E. x E. y ( x = y /\ A = { x } ) <-> E. x A = { x } )
37		eqid 2196	. . . . . . . 8 |- { A } = { A }
38	37	a1bi 243	. . . . . . 7 |- ( E. x E. y ( x = y /\ A = { x } ) <-> ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) )
39	29, 36, 38	3bitr2ri 209	. . . . . 6 |- ( ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) <-> E. w A = { w } )
40	26, 39	sylib 122	. . . . 5 |- ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) -> E. w A = { w } )
41		eqid 2196	. . . . . 6 |- { A , B } = { A , B }
42		prexg 4244	. . . . . . . 8 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
43	4, 5, 42	mp2an 426	. . . . . . 7 |- { A , B } e. _V
44		eqeq1 2203	. . . . . . . 8 |- ( z = { A , B } -> ( z = { A , B } <-> { A , B } = { A , B } ) )
45		eqeq1 2203	. . . . . . . . 9 |- ( z = { A , B } -> ( z = <. x , y >. <-> { A , B } = <. x , y >. ) )
46	45	2exbidv 1882	. . . . . . . 8 |- ( z = { A , B } -> ( E. x E. y z = <. x , y >. <-> E. x E. y { A , B } = <. x , y >. ) )
47	44, 46	imbi12d 234	. . . . . . 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 >. ) ) )
48	43, 47	spcv 2858	. . . . . 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 >. ) )
49	41, 48	mpi 15	. . . . 5 |- ( A. z ( z = { A , B } -> E. x E. y z = <. x , y >. ) -> E. x E. y { A , B } = <. x , y >. )
50		eqcom 2198	. . . . . . . . . 10 |- ( { A , B } = <. x , y >. <-> <. x , y >. = { A , B } )
51	19, 20, 4, 5	opeqpr 4286	. . . . . . . . . 10 |- ( <. x , y >. = { A , B } <-> ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) )
52	50, 51	bitri 184	. . . . . . . . 9 |- ( { A , B } = <. x , y >. <-> ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) )
53		idd 21	. . . . . . . . . 10 |- ( A = { w } -> ( ( A = { x } /\ B = { x , y } ) -> ( A = { x } /\ B = { x , y } ) ) )
54		eqtr2 2215	. . . . . . . . . . . . . 14 |- ( ( A = { x , y } /\ A = { w } ) -> { x , y } = { w } )
55		vex 2766	. . . . . . . . . . . . . . . 16 |- w e. _V
56	19, 20, 55	preqsn 3805	. . . . . . . . . . . . . . 15 |- ( { x , y } = { w } <-> ( x = y /\ y = w ) )
57	56	simplbi 274	. . . . . . . . . . . . . 14 |- ( { x , y } = { w } -> x = y )
58	54, 57	syl 14	. . . . . . . . . . . . 13 |- ( ( A = { x , y } /\ A = { w } ) -> x = y )
59		dfsn2 3636	. . . . . . . . . . . . . . . . . . . 20 |- { x } = { x , x }
60		preq2 3700	. . . . . . . . . . . . . . . . . . . 20 |- ( x = y -> { x , x } = { x , y } )
61	59, 60	eqtr2id 2242	. . . . . . . . . . . . . . . . . . 19 |- ( x = y -> { x , y } = { x } )
62	61	eqeq2d 2208	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( A = { x , y } <-> A = { x } ) )
63	59, 60	eqtrid 2241	. . . . . . . . . . . . . . . . . . 19 |- ( x = y -> { x } = { x , y } )
64	63	eqeq2d 2208	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( B = { x } <-> B = { x , y } ) )
65	62, 64	anbi12d 473	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( ( A = { x , y } /\ B = { x } ) <-> ( A = { x } /\ B = { x , y } ) ) )
66	65	biimpd 144	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( ( A = { x , y } /\ B = { x } ) -> ( A = { x } /\ B = { x , y } ) ) )
67	66	expd 258	. . . . . . . . . . . . . . 15 |- ( x = y -> ( A = { x , y } -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
68	67	com12 30	. . . . . . . . . . . . . 14 |- ( A = { x , y } -> ( x = y -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
69	68	adantr 276	. . . . . . . . . . . . 13 |- ( ( A = { x , y } /\ A = { w } ) -> ( x = y -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
70	58, 69	mpd 13	. . . . . . . . . . . 12 |- ( ( A = { x , y } /\ A = { w } ) -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) )
71	70	expcom 116	. . . . . . . . . . 11 |- ( A = { w } -> ( A = { x , y } -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
72	71	impd 254	. . . . . . . . . 10 |- ( A = { w } -> ( ( A = { x , y } /\ B = { x } ) -> ( A = { x } /\ B = { x , y } ) ) )
73	53, 72	jaod 718	. . . . . . . . 9 |- ( A = { w } -> ( ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) -> ( A = { x } /\ B = { x , y } ) ) )
74	52, 73	biimtrid 152	. . . . . . . 8 |- ( A = { w } -> ( { A , B } = <. x , y >. -> ( A = { x } /\ B = { x , y } ) ) )
75	74	2eximdv 1896	. . . . . . 7 |- ( A = { w } -> ( E. x E. y { A , B } = <. x , y >. -> E. x E. y ( A = { x } /\ B = { x , y } ) ) )
76	75	exlimiv 1612	. . . . . 6 |- ( E. w A = { w } -> ( E. x E. y { A , B } = <. x , y >. -> E. x E. y ( A = { x } /\ B = { x , y } ) ) )
77	76	imp 124	. . . . 5 |- ( ( E. w A = { w } /\ E. x E. y { A , B } = <. x , y >. ) -> E. x E. y ( A = { x } /\ B = { x , y } ) )
78	40, 49, 77	syl2an 289	. . . 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 } ) )
79	14, 78	sylbi 121	. . 3 |- ( <. A , B >. C_ ( _V X. _V ) -> E. x E. y ( A = { x } /\ B = { x , y } ) )
80		simpr 110	. . . . . . . . . . 11 |- ( ( A = { x } /\ z = { A } ) -> z = { A } )
81		equid 1715	. . . . . . . . . . . . . 14 |- x = x
82	81	jctl 314	. . . . . . . . . . . . 13 |- ( A = { x } -> ( x = x /\ A = { x } ) )
83	19, 19, 4	opeqsn 4285	. . . . . . . . . . . . 13 |- ( <. x , x >. = { A } <-> ( x = x /\ A = { x } ) )
84	82, 83	sylibr 134	. . . . . . . . . . . 12 |- ( A = { x } -> <. x , x >. = { A } )
85	84	adantr 276	. . . . . . . . . . 11 |- ( ( A = { x } /\ z = { A } ) -> <. x , x >. = { A } )
86	80, 85	eqtr4d 2232	. . . . . . . . . 10 |- ( ( A = { x } /\ z = { A } ) -> z = <. x , x >. )
87		opeq12 3810	. . . . . . . . . . . 12 |- ( ( w = x /\ v = x ) -> <. w , v >. = <. x , x >. )
88	87	eqeq2d 2208	. . . . . . . . . . 11 |- ( ( w = x /\ v = x ) -> ( z = <. w , v >. <-> z = <. x , x >. ) )
89	19, 19, 88	spc2ev 2860	. . . . . . . . . 10 |- ( z = <. x , x >. -> E. w E. v z = <. w , v >. )
90	86, 89	syl 14	. . . . . . . . 9 |- ( ( A = { x } /\ z = { A } ) -> E. w E. v z = <. w , v >. )
91	90	adantlr 477	. . . . . . . 8 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A } ) -> E. w E. v z = <. w , v >. )
92		preq12 3701	. . . . . . . . . . . 12 |- ( ( A = { x } /\ B = { x , y } ) -> { A , B } = { { x } , { x , y } } )
93	92	eqeq2d 2208	. . . . . . . . . . 11 |- ( ( A = { x } /\ B = { x , y } ) -> ( z = { A , B } <-> z = { { x } , { x , y } } ) )
94	93	biimpa 296	. . . . . . . . . 10 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> z = { { x } , { x , y } } )
95	19, 20	dfop 3807	. . . . . . . . . 10 |- <. x , y >. = { { x } , { x , y } }
96	94, 95	eqtr4di 2247	. . . . . . . . 9 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> z = <. x , y >. )
97		opeq12 3810	. . . . . . . . . . 11 |- ( ( w = x /\ v = y ) -> <. w , v >. = <. x , y >. )
98	97	eqeq2d 2208	. . . . . . . . . 10 |- ( ( w = x /\ v = y ) -> ( z = <. w , v >. <-> z = <. x , y >. ) )
99	19, 20, 98	spc2ev 2860	. . . . . . . . 9 |- ( z = <. x , y >. -> E. w E. v z = <. w , v >. )
100	96, 99	syl 14	. . . . . . . 8 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> E. w E. v z = <. w , v >. )
101	91, 100	jaodan 798	. . . . . . 7 |- ( ( ( A = { x } /\ B = { x , y } ) /\ ( z = { A } \/ z = { A , B } ) ) -> E. w E. v z = <. w , v >. )
102	101	ex 115	. . . . . 6 |- ( ( A = { x } /\ B = { x , y } ) -> ( ( z = { A } \/ z = { A , B } ) -> E. w E. v z = <. w , v >. ) )
103		elvv 4725	. . . . . 6 |- ( z e. ( _V X. _V ) <-> E. w E. v z = <. w , v >. )
104	102, 6, 103	3imtr4g 205	. . . . 5 |- ( ( A = { x } /\ B = { x , y } ) -> ( z e. <. A , B >. -> z e. ( _V X. _V ) ) )
105	104	ssrdv 3189	. . . 4 |- ( ( A = { x } /\ B = { x , y } ) -> <. A , B >. C_ ( _V X. _V ) )
106	105	exlimivv 1911	. . 3 |- ( E. x E. y ( A = { x } /\ B = { x , y } ) -> <. A , B >. C_ ( _V X. _V ) )
107	79, 106	impbii 126	. 2 |- ( <. A , B >. C_ ( _V X. _V ) <-> E. x E. y ( A = { x } /\ B = { x , y } ) )
108	1, 107	bitri 184	1 |- ( Rel <. A , B >. <-> E. x E. y ( A = { x } /\ B = { x , y } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   A.wal 1362   = wceq 1364   E.wex 1506   e. wcel 2167   _Vcvv 2763   C_ wss 3157   {csn 3622   {cpr 3623   <.cop 3625   X. cxp 4661   Rel wrel 4668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-xp 4669  df-rel 4670

This theorem is referenced by:  funopg  5292