Theorem relop 4990

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 4846	. 2 |- ( Rel <. A , B >. <-> <. A , B >. C_ ( _V X. _V ) )
2		dfss2 3407	. . . . 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 4667	. . . . . . . . 9 |- ( z e. <. A , B >. <-> ( z = { A } \/ z = { A , B } ) )
6		elvv 4898	. . . . . . . . 9 |- ( z e. ( _V X. _V ) <-> E. x E. y z = <. x , y >. )
7	5, 6	imbi12i 333	. . . . . . . 8 |- ( ( z e. <. A , B >. -> z e. ( _V X. _V ) ) <-> ( ( z = { A } \/ z = { A , B } ) -> E. x E. y z = <. x , y >. ) )
8		jaob 800	. . . . . . . 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 257	. . . . . . 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 1699	. . . . . 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 1740	. . . . . 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 257	. . . . 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 257	. . . 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 4641	. . . . . . 7 |- { A } e. _V
15		eqeq1 2475	. . . . . . . 8 |- ( z = { A } -> ( z = { A } <-> { A } = { A } ) )
16		eqeq1 2475	. . . . . . . . . 10 |- ( z = { A } -> ( z = <. x , y >. <-> { A } = <. x , y >. ) )
17		eqcom 2478	. . . . . . . . . . 11 |- ( { A } = <. x , y >. <-> <. x , y >. = { A } )
18		vex 3034	. . . . . . . . . . . 12 |- x e. _V
19		vex 3034	. . . . . . . . . . . 12 |- y e. _V
20	18, 19, 3	opeqsn 4697	. . . . . . . . . . 11 |- ( <. x , y >. = { A } <-> ( x = y /\ A = { x } ) )
21	17, 20	bitri 257	. . . . . . . . . 10 |- ( { A } = <. x , y >. <-> ( x = y /\ A = { x } ) )
22	16, 21	syl6bb 269	. . . . . . . . 9 |- ( z = { A } -> ( z = <. x , y >. <-> ( x = y /\ A = { x } ) ) )
23	22	2exbidv 1778	. . . . . . . 8 |- ( z = { A } -> ( E. x E. y z = <. x , y >. <-> E. x E. y ( x = y /\ A = { x } ) ) )
24	15, 23	imbi12d 327	. . . . . . 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 3126	. . . . . 6 |- ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) -> ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) )
26		sneq 3969	. . . . . . . . 9 |- ( w = x -> { w } = { x } )
27	26	eqeq2d 2481	. . . . . . . 8 |- ( w = x -> ( A = { w } <-> A = { x } ) )
28	27	cbvexv 2130	. . . . . . 7 |- ( E. w A = { w } <-> E. x A = { x } )
29		ax6evr 1867	. . . . . . . . 9 |- E. y x = y
30		19.41v 1838	. . . . . . . . 9 |- ( E. y ( x = y /\ A = { x } ) <-> ( E. y x = y /\ A = { x } ) )
31	29, 30	mpbiran 932	. . . . . . . 8 |- ( E. y ( x = y /\ A = { x } ) <-> A = { x } )
32	31	exbii 1726	. . . . . . 7 |- ( E. x E. y ( x = y /\ A = { x } ) <-> E. x A = { x } )
33		eqid 2471	. . . . . . . 8 |- { A } = { A }
34	33	a1bi 344	. . . . . . 7 |- ( E. x E. y ( x = y /\ A = { x } ) <-> ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) )
35	28, 32, 34	3bitr2ri 282	. . . . . 6 |- ( ( { A } = { A } -> E. x E. y ( x = y /\ A = { x } ) ) <-> E. w A = { w } )
36	25, 35	sylib 201	. . . . 5 |- ( A. z ( z = { A } -> E. x E. y z = <. x , y >. ) -> E. w A = { w } )
37		eqid 2471	. . . . . 6 |- { A , B } = { A , B }
38		prex 4642	. . . . . . 7 |- { A , B } e. _V
39		eqeq1 2475	. . . . . . . 8 |- ( z = { A , B } -> ( z = { A , B } <-> { A , B } = { A , B } ) )
40		eqeq1 2475	. . . . . . . . 9 |- ( z = { A , B } -> ( z = <. x , y >. <-> { A , B } = <. x , y >. ) )
41	40	2exbidv 1778	. . . . . . . 8 |- ( z = { A , B } -> ( E. x E. y z = <. x , y >. <-> E. x E. y { A , B } = <. x , y >. ) )
42	39, 41	imbi12d 327	. . . . . . 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 3126	. . . . . 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 2478	. . . . . . . . . 10 |- ( { A , B } = <. x , y >. <-> <. x , y >. = { A , B } )
46	18, 19, 3, 4	opeqpr 4698	. . . . . . . . . 10 |- ( <. x , y >. = { A , B } <-> ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) )
47	45, 46	bitri 257	. . . . . . . . 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 2491	. . . . . . . . . . . . . 14 |- ( ( A = { x , y } /\ A = { w } ) -> { x , y } = { w } )
50		vex 3034	. . . . . . . . . . . . . . . 16 |- w e. _V
51	18, 19, 50	preqsn 4151	. . . . . . . . . . . . . . 15 |- ( { x , y } = { w } <-> ( x = y /\ y = w ) )
52	51	simplbi 467	. . . . . . . . . . . . . 14 |- ( { x , y } = { w } -> x = y )
53	49, 52	syl 17	. . . . . . . . . . . . 13 |- ( ( A = { x , y } /\ A = { w } ) -> x = y )
54		dfsn2 3972	. . . . . . . . . . . . . . . . . . . 20 |- { x } = { x , x }
55		preq2 4043	. . . . . . . . . . . . . . . . . . . 20 |- ( x = y -> { x , x } = { x , y } )
56	54, 55	syl5req 2518	. . . . . . . . . . . . . . . . . . 19 |- ( x = y -> { x , y } = { x } )
57	56	eqeq2d 2481	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( A = { x , y } <-> A = { x } ) )
58	54, 55	syl5eq 2517	. . . . . . . . . . . . . . . . . . 19 |- ( x = y -> { x } = { x , y } )
59	58	eqeq2d 2481	. . . . . . . . . . . . . . . . . 18 |- ( x = y -> ( B = { x } <-> B = { x , y } ) )
60	57, 59	anbi12d 725	. . . . . . . . . . . . . . . . 17 |- ( x = y -> ( ( A = { x , y } /\ B = { x } ) <-> ( A = { x } /\ B = { x , y } ) ) )
61	60	biimpd 212	. . . . . . . . . . . . . . . 16 |- ( x = y -> ( ( A = { x , y } /\ B = { x } ) -> ( A = { x } /\ B = { x , y } ) ) )
62	61	expd 443	. . . . . . . . . . . . . . 15 |- ( x = y -> ( A = { x , y } -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
63	62	com12 31	. . . . . . . . . . . . . 14 |- ( A = { x , y } -> ( x = y -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
64	63	adantr 472	. . . . . . . . . . . . 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 442	. . . . . . . . . . 11 |- ( A = { w } -> ( A = { x , y } -> ( B = { x } -> ( A = { x } /\ B = { x , y } ) ) ) )
67	66	impd 438	. . . . . . . . . 10 |- ( A = { w } -> ( ( A = { x , y } /\ B = { x } ) -> ( A = { x } /\ B = { x , y } ) ) )
68	48, 67	jaod 387	. . . . . . . . 9 |- ( A = { w } -> ( ( ( A = { x } /\ B = { x , y } ) \/ ( A = { x , y } /\ B = { x } ) ) -> ( A = { x } /\ B = { x , y } ) ) )
69	47, 68	syl5bi 225	. . . . . . . 8 |- ( A = { w } -> ( { A , B } = <. x , y >. -> ( A = { x } /\ B = { x , y } ) ) )
70	69	2eximdv 1774	. . . . . . 7 |- ( A = { w } -> ( E. x E. y { A , B } = <. x , y >. -> E. x E. y ( A = { x } /\ B = { x , y } ) ) )
71	70	exlimiv 1784	. . . . . 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 436	. . . . 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 485	. . . 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 200	. . 3 |- ( <. A , B >. C_ ( _V X. _V ) -> E. x E. y ( A = { x } /\ B = { x , y } ) )
75		simpr 468	. . . . . . . . . . 11 |- ( ( A = { x } /\ z = { A } ) -> z = { A } )
76		equid 1863	. . . . . . . . . . . . . 14 |- x = x
77	76	jctl 550	. . . . . . . . . . . . 13 |- ( A = { x } -> ( x = x /\ A = { x } ) )
78	18, 18, 3	opeqsn 4697	. . . . . . . . . . . . 13 |- ( <. x , x >. = { A } <-> ( x = x /\ A = { x } ) )
79	77, 78	sylibr 217	. . . . . . . . . . . 12 |- ( A = { x } -> <. x , x >. = { A } )
80	79	adantr 472	. . . . . . . . . . 11 |- ( ( A = { x } /\ z = { A } ) -> <. x , x >. = { A } )
81	75, 80	eqtr4d 2508	. . . . . . . . . 10 |- ( ( A = { x } /\ z = { A } ) -> z = <. x , x >. )
82		opeq12 4160	. . . . . . . . . . . 12 |- ( ( w = x /\ v = x ) -> <. w , v >. = <. x , x >. )
83	82	eqeq2d 2481	. . . . . . . . . . 11 |- ( ( w = x /\ v = x ) -> ( z = <. w , v >. <-> z = <. x , x >. ) )
84	18, 18, 83	spc2ev 3128	. . . . . . . . . 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 729	. . . . . . . 8 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A } ) -> E. w E. v z = <. w , v >. )
87		preq12 4044	. . . . . . . . . . . 12 |- ( ( A = { x } /\ B = { x , y } ) -> { A , B } = { { x } , { x , y } } )
88	87	eqeq2d 2481	. . . . . . . . . . 11 |- ( ( A = { x } /\ B = { x , y } ) -> ( z = { A , B } <-> z = { { x } , { x , y } } ) )
89	88	biimpa 492	. . . . . . . . . 10 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> z = { { x } , { x , y } } )
90	18, 19	dfop 4157	. . . . . . . . . 10 |- <. x , y >. = { { x } , { x , y } }
91	89, 90	syl6eqr 2523	. . . . . . . . 9 |- ( ( ( A = { x } /\ B = { x , y } ) /\ z = { A , B } ) -> z = <. x , y >. )
92		opeq12 4160	. . . . . . . . . . 11 |- ( ( w = x /\ v = y ) -> <. w , v >. = <. x , y >. )
93	92	eqeq2d 2481	. . . . . . . . . 10 |- ( ( w = x /\ v = y ) -> ( z = <. w , v >. <-> z = <. x , y >. ) )
94	18, 19, 93	spc2ev 3128	. . . . . . . . 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 802	. . . . . . 7 |- ( ( ( A = { x } /\ B = { x , y } ) /\ ( z = { A } \/ z = { A , B } ) ) -> E. w E. v z = <. w , v >. )
97	96	ex 441	. . . . . 6 |- ( ( A = { x } /\ B = { x , y } ) -> ( ( z = { A } \/ z = { A , B } ) -> E. w E. v z = <. w , v >. ) )
98		elvv 4898	. . . . . 6 |- ( z e. ( _V X. _V ) <-> E. w E. v z = <. w , v >. )
99	97, 5, 98	3imtr4g 278	. . . . 5 |- ( ( A = { x } /\ B = { x , y } ) -> ( z e. <. A , B >. -> z e. ( _V X. _V ) ) )
100	99	ssrdv 3424	. . . 4 |- ( ( A = { x } /\ B = { x , y } ) -> <. A , B >. C_ ( _V X. _V ) )
101	100	exlimivv 1786	. . 3 |- ( E. x E. y ( A = { x } /\ B = { x , y } ) -> <. A , B >. C_ ( _V X. _V ) )
102	74, 101	impbii 192	. 2 |- ( <. A , B >. C_ ( _V X. _V ) <-> E. x E. y ( A = { x } /\ B = { x , y } ) )
103	1, 102	bitri 257	1 |- ( Rel <. A , B >. <-> E. x E. y ( A = { x } /\ B = { x , y } ) )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 375   /\ wa 376   A.wal 1450   = wceq 1452   E.wex 1671   e. wcel 1904   _Vcvv 3031   C_ wss 3390   {csn 3959   {cpr 3961   <.cop 3965   X. cxp 4837   Rel wrel 4844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845  df-rel 4846

This theorem is referenced by:  funopg  5621