Theorem relop 5146

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 5001	. 2 |- ( Rel <. A , B >. <-> <. A , B >. C_ ( _V X. _V ) )
2		dfss2 3488	. . . . 5 |- ( <. A , B >. C_ ( _V X. _V ) <-> A. z ( z e. <. A , B >. -> z e. ( _V X. _V ) ) )
3		vex 3111	. . . . . . . . . 10 |- z e. _V
4		relop.1	. . . . . . . . . 10 |- A e. _V
5		relop.2	. . . . . . . . . 10 |- B e. _V
6	3, 4, 5	elop 4708	. . . . . . . . 9 |- ( z e. <. A , B >. <-> ( z = { A } \/ z = { A , B } ) )
7		elvv 5052	. . . . . . . . 9 |- ( z e. ( _V X. _V ) <-> E. x E. y z = <. x , y >. )
8	6, 7	imbi12i 326	. . . . . . . 8 |- ( ( z e. <. A , B >. -> z e. ( _V X. _V ) ) <-> ( ( z = { A } \/ z = { A , B } ) -> E. x E. y z = <. x , y >. ) )
9		jaob 781	. . . . . . . 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 249	. . . . . . 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 1615	. . . . . 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 1652	. . . . . 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 249	. . . . 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 249	. . . 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		snex 4683	. . . . . . 7 |- { A } e. _V
16		eqeq1 2466	. . . . . . . 8 |- ( z = { A } -> ( z = { A } <-> { A } = { A } ) )
17		eqeq1 2466	. . . . . . . . . 10 |- ( z = { A } -> ( z = <. x , y >. <-> { A } = <. x , y >. ) )
18		eqcom 2471	. . . . . . . . . . 11 |- ( { A } = <. x , y >. <-> <. x , y >. = { A } )
19		vex 3111	. . . . . . . . . . . 12 |- x e. _V
20		vex 3111	. . . . . . . . . . . 12 |- y e. _V
21	19, 20, 4	opeqsn 4738	. . . . . . . . . . 11 |- ( <. x , y >. = { A } <-> ( x = y /\ A = { x } ) )
22	18, 21	bitri 249	. . . . . . . . . 10 |- ( { A } = <. x , y >. <-> ( x = y /\ A = { x } ) )
23	17, 22	syl6bb 261	. . . . . . . . 9 |- ( z = { A } -> ( z = <. x , y >. <-> ( x = y /\ A = { x } ) ) )
24	23	2exbidv 1687	. . . . . . . 8 |- ( z = { A } -> ( E. x E. y z = <. x , y >. <-> E. x E. y ( x = y /\ A = { x } ) ) )
25	16, 24	imbi12d 320	. . . . . . 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 3199	. . . . . 6 |- ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) -> ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) )
27		sneq 4032	. . . . . . . . 9 |- ( w = x -> { w } = { x } )
28	27	eqeq2d 2476	. . . . . . . 8 |- ( w = x -> ( A = { w } <-> A = { x } ) )
29	28	cbvexv 1992	. . . . . . 7 |- ( E. w A = { w } <-> E. x A = { x } )
30		ax6ev 1716	. . . . . . . . . 10 |- E. y y = x
31		equcom 1738	. . . . . . . . . . 11 |- ( y = x <-> x = y )
32	31	exbii 1639	. . . . . . . . . 10 |- ( E. y y = x <-> E. y x = y )
33	30, 32	mpbi 208	. . . . . . . . 9 |- E. y x = y
34		19.41v 1940	. . . . . . . . 9 |- ( E. y ( x = y /\ A = { x } ) <-> ( E. y x = y /\ A = { x } ) )
35	33, 34	mpbiran 911	. . . . . . . 8 |- ( E. y ( x = y /\ A = { x } ) <-> A = { x } )
36	35	exbii 1639	. . . . . . 7 |- ( E. x E. y ( x = y /\ A = { x } ) <-> E. x A = { x } )
37		eqid 2462	. . . . . . . 8 |- { A } = { A }
38	37	a1bi 337	. . . . . . 7 |- ( E. x E. y ( x = y /\ A = { x } ) <-> ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) )
39	29, 36, 38	3bitr2ri 274	. . . . . 6 |- ( ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) <-> E. w A = { w } )
40	26, 39	sylib 196	. . . . 5 |- ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) -> E. w A = { w } )
41		eqid 2462	. . . . . 6 |- { A , B } = { A , B }
42		prex 4684	. . . . . . 7 |- { A , B } e. _V
43		eqeq1 2466	. . . . . . . 8 |- ( z = { A , B } -> ( z = { A , B } <-> { A , B } = { A , B } ) )
44		eqeq1 2466	. . . . . . . . 9 |- ( z = { A , B } -> ( z = <. x , y >. <-> { A , B } = <. x , y >. ) )
45	44	2exbidv 1687	. . . . . . . 8 |- ( z = { A , B } -> ( E. x E. y z = <. x , y >. <-> E. x E. y { A , B } = <. x , y >. ) )
46	43, 45	imbi12d 320	. . . . . . 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 >. ) ) )
47	42, 46	spcv 3199	. . . . . 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 >. ) )
48	41, 47	mpi 17	. . . . 5 |- ( A. z ( z = { A , B } -> E. x E. y z = <. x , y >. ) -> E. x E. y { A , B } = <. x , y >. )
49		eqcom 2471	. . . . . . . . . 10 |- ( { A , B } = <. x , y >. <-> <. x , y >. = { A , B } )
50	19, 20, 4, 5	opeqpr 4739	. . . . . . . . . 10 |- ( <. x , y >. = { A , B } <-> ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) )
51	49, 50	bitri 249	. . . . . . . . 9 |- ( { A , B } = <. x , y >. <-> ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) )
52		idd 24	. . . . . . . . . 10 |- ( A = { w } -> ( ( A = { x } /\ B = { x , y } ) -> ( A = { x } /\ B = { x , y } ) ) )
53		eqtr2 2489	. . . . . . . . . . . . . 14 |- ( ( A = { x , y } /\ A = { w } ) -> { x , y } = { w } )
54		vex 3111	. . . . . . . . . . . . . . . 16 |- w e. _V
55	19, 20, 54	preqsn 4204	. . . . . . . . . . . . . . 15 |- ( { x , y } = { w } <-> ( x = y /\ y = w ) )
56	55	simplbi 460	. . . . . . . . . . . . . 14 |- ( { x , y } = { w } -> x = y )
57	53, 56	syl 16	. . . . . . . . . . . . 13 |- ( ( A = { x , y } /\ A = { w } ) -> x = y )
58		dfsn2 4035	. . . . . . . . . . . . . . . . . . . 20 |- { x } = { x , x }
59		preq2 4102	. . . . . . . . . . . . . . . . . . . 20 |- ( x = y -> { x , x } = { x , y } )
60	58, 59	syl5req 2516	. . . . . . . . . . . . . . . . . . 19 |- ( x = y -> { x , y } = { x } )
61	60	eqeq2d 2476	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( A = { x , y } <-> A = { x } ) )
62	58, 59	syl5eq 2515	. . . . . . . . . . . . . . . . . . 19 |- ( x = y -> { x } = { x , y } )
63	62	eqeq2d 2476	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( B = { x } <-> B = { x , y } ) )
64	61, 63	anbi12d 710	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( ( A = { x , y } /\ B = { x } ) <-> ( A = { x } /\ B = { x , y } ) ) )
65	64	biimpd 207	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( ( A = { x , y } /\ B = { x } ) -> ( A = { x } /\ B = { x , y } ) ) )
66	65	expd 436	. . . . . . . . . . . . . . 15 |- ( x = y -> ( A = { x , y } -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
67	66	com12 31	. . . . . . . . . . . . . 14 |- ( A = { x , y } -> ( x = y -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
68	67	adantr 465	. . . . . . . . . . . . 13 |- ( ( A = { x , y } /\ A = { w } ) -> ( x = y -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
69	57, 68	mpd 15	. . . . . . . . . . . 12 |- ( ( A = { x , y } /\ A = { w } ) -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) )
70	69	expcom 435	. . . . . . . . . . 11 |- ( A = { w } -> ( A = { x , y } -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
71	70	impd 431	. . . . . . . . . 10 |- ( A = { w } -> ( ( A = { x , y } /\ B = { x } ) -> ( A = { x } /\ B = { x , y } ) ) )
72	52, 71	jaod 380	. . . . . . . . 9 |- ( A = { w } -> ( ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) -> ( A = { x } /\ B = { x , y } ) ) )
73	51, 72	syl5bi 217	. . . . . . . 8 |- ( A = { w } -> ( { A , B } = <. x , y >. -> ( A = { x } /\ B = { x , y } ) ) )
74	73	2eximdv 1683	. . . . . . 7 |- ( A = { w } -> ( E. x E. y { A , B } = <. x , y >. -> E. x E. y ( A = { x } /\ B = { x , y } ) ) )
75	74	exlimiv 1693	. . . . . 6 |- ( E. w A = { w } -> ( E. x E. y { A , B } = <. x , y >. -> E. x E. y ( A = { x } /\ B = { x , y } ) ) )
76	75	imp 429	. . . . 5 |- ( ( E. w A = { w } /\ E. x E. y { A , B } = <. x , y >. ) -> E. x E. y ( A = { x } /\ B = { x , y } ) )
77	40, 48, 76	syl2an 477	. . . 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 } ) )
78	14, 77	sylbi 195	. . 3 |- ( <. A , B >. C_ ( _V X. _V ) -> E. x E. y ( A = { x } /\ B = { x , y } ) )
79		simpr 461	. . . . . . . . . . 11 |- ( ( A = { x } /\ z = { A } ) -> z = { A } )
80		equid 1735	. . . . . . . . . . . . . 14 |- x = x
81	80	jctl 541	. . . . . . . . . . . . 13 |- ( A = { x } -> ( x = x /\ A = { x } ) )
82	19, 19, 4	opeqsn 4738	. . . . . . . . . . . . 13 |- ( <. x , x >. = { A } <-> ( x = x /\ A = { x } ) )
83	81, 82	sylibr 212	. . . . . . . . . . . 12 |- ( A = { x } -> <. x , x >. = { A } )
84	83	adantr 465	. . . . . . . . . . 11 |- ( ( A = { x } /\ z = { A } ) -> <. x , x >. = { A } )
85	79, 84	eqtr4d 2506	. . . . . . . . . 10 |- ( ( A = { x } /\ z = { A } ) -> z = <. x , x >. )
86		opeq12 4210	. . . . . . . . . . . 12 |- ( ( w = x /\ v = x ) -> <. w , v >. = <. x , x >. )
87	86	eqeq2d 2476	. . . . . . . . . . 11 |- ( ( w = x /\ v = x ) -> ( z = <. w , v >. <-> z = <. x , x >. ) )
88	19, 19, 87	spc2ev 3201	. . . . . . . . . 10 |- ( z = <. x , x >. -> E. w E. v z = <. w , v >. )
89	85, 88	syl 16	. . . . . . . . 9 |- ( ( A = { x } /\ z = { A } ) -> E. w E. v z = <. w , v >. )
90	89	adantlr 714	. . . . . . . 8 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A } ) -> E. w E. v z = <. w , v >. )
91		preq12 4103	. . . . . . . . . . . 12 |- ( ( A = { x } /\ B = { x , y } ) -> { A , B } = { { x } , { x , y } } )
92	91	eqeq2d 2476	. . . . . . . . . . 11 |- ( ( A = { x } /\ B = { x , y } ) -> ( z = { A , B } <-> z = { { x } , { x , y } } ) )
93	92	biimpa 484	. . . . . . . . . 10 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> z = { { x } , { x , y } } )
94	19, 20	dfop 4207	. . . . . . . . . 10 |- <. x , y >. = { { x } , { x , y } }
95	93, 94	syl6eqr 2521	. . . . . . . . 9 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> z = <. x , y >. )
96		opeq12 4210	. . . . . . . . . . 11 |- ( ( w = x /\ v = y ) -> <. w , v >. = <. x , y >. )
97	96	eqeq2d 2476	. . . . . . . . . 10 |- ( ( w = x /\ v = y ) -> ( z = <. w , v >. <-> z = <. x , y >. ) )
98	19, 20, 97	spc2ev 3201	. . . . . . . . 9 |- ( z = <. x , y >. -> E. w E. v z = <. w , v >. )
99	95, 98	syl 16	. . . . . . . 8 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> E. w E. v z = <. w , v >. )
100	90, 99	jaodan 783	. . . . . . 7 |- ( ( ( A = { x } /\ B = { x , y } ) /\ ( z = { A } \/ z = { A , B } ) ) -> E. w E. v z = <. w , v >. )
101	100	ex 434	. . . . . 6 |- ( ( A = { x } /\ B = { x , y } ) -> ( ( z = { A } \/ z = { A , B } ) -> E. w E. v z = <. w , v >. ) )
102		elvv 5052	. . . . . 6 |- ( z e. ( _V X. _V ) <-> E. w E. v z = <. w , v >. )
103	101, 6, 102	3imtr4g 270	. . . . 5 |- ( ( A = { x } /\ B = { x , y } ) -> ( z e. <. A , B >. -> z e. ( _V X. _V ) ) )
104	103	ssrdv 3505	. . . 4 |- ( ( A = { x } /\ B = { x , y } ) -> <. A , B >. C_ ( _V X. _V ) )
105	104	exlimivv 1694	. . 3 |- ( E. x E. y ( A = { x } /\ B = { x , y } ) -> <. A , B >. C_ ( _V X. _V ) )
106	78, 105	impbii 188	. 2 |- ( <. A , B >. C_ ( _V X. _V ) <-> E. x E. y ( A = { x } /\ B = { x , y } ) )
107	1, 106	bitri 249	1 |- ( Rel <. A , B >. <-> E. x E. y ( A = { x } /\ B = { x , y } ) )

Syntax hints:   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   A.wal 1372   = wceq 1374   E.wex 1591   e. wcel 1762   _Vcvv 3108   C_ wss 3471   {csn 4022   {cpr 4024   <.cop 4028   X. cxp 4992   Rel wrel 4999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-opab 4501  df-xp 5000  df-rel 5001

This theorem is referenced by:  funopg  5613