Theorem funopg 4934

Description: A Kuratowski ordered pair is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
funopg	|- ( ( A e. V /\ B e. W /\ Fun <. A , B >. ) -> A = B )

Proof of Theorem funopg

Dummy variables u t v w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3549	. . . . 5 |- ( u = A -> <. u , t >. = <. A , t >. )
2	1	funeqd 4923	. . . 4 |- ( u = A -> ( Fun <. u , t >. <-> Fun <. A , t >. ) )
3		eqeq1 2046	. . . 4 |- ( u = A -> ( u = t <-> A = t ) )
4	2, 3	imbi12d 223	. . 3 |- ( u = A -> ( ( Fun <. u , t >. -> u = t ) <-> ( Fun <. A , t >. -> A = t ) ) )
5		opeq2 3550	. . . . 5 |- ( t = B -> <. A , t >. = <. A , B >. )
6	5	funeqd 4923	. . . 4 |- ( t = B -> ( Fun <. A , t >. <-> Fun <. A , B >. ) )
7		eqeq2 2049	. . . 4 |- ( t = B -> ( A = t <-> A = B ) )
8	6, 7	imbi12d 223	. . 3 |- ( t = B -> ( ( Fun <. A , t >. -> A = t ) <-> ( Fun <. A , B >. -> A = B ) ) )
9		funrel 4919	. . . . 5 |- ( Fun <. u , t >. -> Rel <. u , t >. )
10		vex 2560	. . . . . 6 |- u e. _V
11		vex 2560	. . . . . 6 |- t e. _V
12	10, 11	relop 4486	. . . . 5 |- ( Rel <. u , t >. <-> E. x E. y ( u = { x } /\ t = { x , y } ) )
13	9, 12	sylib 127	. . . 4 |- ( Fun <. u , t >. -> E. x E. y ( u = { x } /\ t = { x , y } ) )
14	10, 11	opth 3974	. . . . . . . 8 |- ( <. u , t >. = <. { x } , { x , y } >. <-> ( u = { x } /\ t = { x , y } ) )
15		vex 2560	. . . . . . . . . . . 12 |- x e. _V
16	15	opid 3567	. . . . . . . . . . 11 |- <. x , x >. = { { x } }
17	16	preq1i 3450	. . . . . . . . . 10 |- { <. x , x >. , { { x } , { x , y } } } = { { { x } } , { { x } , { x , y } } }
18		vex 2560	. . . . . . . . . . . 12 |- y e. _V
19	15, 18	dfop 3548	. . . . . . . . . . 11 |- <. x , y >. = { { x } , { x , y } }
20	19	preq2i 3451	. . . . . . . . . 10 |- { <. x , x >. , <. x , y >. } = { <. x , x >. , { { x } , { x , y } } }
21		snexgOLD 3935	. . . . . . . . . . . 12 |- ( x e. _V -> { x } e. _V )
22	15, 21	ax-mp 7	. . . . . . . . . . 11 |- { x } e. _V
23		zfpair2 3945	. . . . . . . . . . 11 |- { x , y } e. _V
24	22, 23	dfop 3548	. . . . . . . . . 10 |- <. { x } , { x , y } >. = { { { x } } , { { x } , { x , y } } }
25	17, 20, 24	3eqtr4ri 2071	. . . . . . . . 9 |- <. { x } , { x , y } >. = { <. x , x >. , <. x , y >. }
26	25	eqeq2i 2050	. . . . . . . 8 |- ( <. u , t >. = <. { x } , { x , y } >. <-> <. u , t >. = { <. x , x >. , <. x , y >. } )
27	14, 26	bitr3i 175	. . . . . . 7 |- ( ( u = { x } /\ t = { x , y } ) <-> <. u , t >. = { <. x , x >. , <. x , y >. } )
28		dffun4 4913	. . . . . . . . 9 |- ( Fun <. u , t >. <-> ( Rel <. u , t >. /\ A. z A. w A. v ( ( <. z , w >. e. <. u , t >. /\ <. z , v >. e. <. u , t >. ) -> w = v ) ) )
29	28	simprbi 260	. . . . . . . 8 |- ( Fun <. u , t >. -> A. z A. w A. v ( ( <. z , w >. e. <. u , t >. /\ <. z , v >. e. <. u , t >. ) -> w = v ) )
30	15, 15	opex 3966	. . . . . . . . . . 11 |- <. x , x >. e. _V
31	30	prid1 3476	. . . . . . . . . 10 |- <. x , x >. e. { <. x , x >. , <. x , y >. }
32		eleq2 2101	. . . . . . . . . 10 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> ( <. x , x >. e. <. u , t >. <-> <. x , x >. e. { <. x , x >. , <. x , y >. } ) )
33	31, 32	mpbiri 157	. . . . . . . . 9 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> <. x , x >. e. <. u , t >. )
34	15, 18	opex 3966	. . . . . . . . . . 11 |- <. x , y >. e. _V
35	34	prid2 3477	. . . . . . . . . 10 |- <. x , y >. e. { <. x , x >. , <. x , y >. }
36		eleq2 2101	. . . . . . . . . 10 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> ( <. x , y >. e. <. u , t >. <-> <. x , y >. e. { <. x , x >. , <. x , y >. } ) )
37	35, 36	mpbiri 157	. . . . . . . . 9 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> <. x , y >. e. <. u , t >. )
38	33, 37	jca 290	. . . . . . . 8 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> ( <. x , x >. e. <. u , t >. /\ <. x , y >. e. <. u , t >. ) )
39		opeq12 3551	. . . . . . . . . . . . . 14 |- ( ( z = x /\ w = x ) -> <. z , w >. = <. x , x >. )
40	39	3adant3 924	. . . . . . . . . . . . 13 |- ( ( z = x /\ w = x /\ v = y ) -> <. z , w >. = <. x , x >. )
41	40	eleq1d 2106	. . . . . . . . . . . 12 |- ( ( z = x /\ w = x /\ v = y ) -> ( <. z , w >. e. <. u , t >. <-> <. x , x >. e. <. u , t >. ) )
42		opeq12 3551	. . . . . . . . . . . . . 14 |- ( ( z = x /\ v = y ) -> <. z , v >. = <. x , y >. )
43	42	3adant2 923	. . . . . . . . . . . . 13 |- ( ( z = x /\ w = x /\ v = y ) -> <. z , v >. = <. x , y >. )
44	43	eleq1d 2106	. . . . . . . . . . . 12 |- ( ( z = x /\ w = x /\ v = y ) -> ( <. z , v >. e. <. u , t >. <-> <. x , y >. e. <. u , t >. ) )
45	41, 44	anbi12d 442	. . . . . . . . . . 11 |- ( ( z = x /\ w = x /\ v = y ) -> ( ( <. z , w >. e. <. u , t >. /\ <. z , v >. e. <. u , t >. ) <-> ( <. x , x >. e. <. u , t >. /\ <. x , y >. e. <. u , t >. ) ) )
46		eqeq12 2052	. . . . . . . . . . . 12 |- ( ( w = x /\ v = y ) -> ( w = v <-> x = y ) )
47	46	3adant1 922	. . . . . . . . . . 11 |- ( ( z = x /\ w = x /\ v = y ) -> ( w = v <-> x = y ) )
48	45, 47	imbi12d 223	. . . . . . . . . 10 |- ( ( z = x /\ w = x /\ v = y ) -> ( ( ( <. z , w >. e. <. u , t >. /\ <. z , v >. e. <. u , t >. ) -> w = v ) <-> ( ( <. x , x >. e. <. u , t >. /\ <. x , y >. e. <. u , t >. ) -> x = y ) ) )
49	48	spc3gv 2645	. . . . . . . . 9 |- ( ( x e. _V /\ x e. _V /\ y e. _V ) -> ( A. z A. w A. v ( ( <. z , w >. e. <. u , t >. /\ <. z , v >. e. <. u , t >. ) -> w = v ) -> ( ( <. x , x >. e. <. u , t >. /\ <. x , y >. e. <. u , t >. ) -> x = y ) ) )
50	15, 15, 18, 49	mp3an 1232	. . . . . . . 8 |- ( A. z A. w A. v ( ( <. z , w >. e. <. u , t >. /\ <. z , v >. e. <. u , t >. ) -> w = v ) -> ( ( <. x , x >. e. <. u , t >. /\ <. x , y >. e. <. u , t >. ) -> x = y ) )
51	29, 38, 50	syl2im 34	. . . . . . 7 |- ( Fun <. u , t >. -> ( <. u , t >. = { <. x , x >. , <. x , y >. } -> x = y ) )
52	27, 51	syl5bi 141	. . . . . 6 |- ( Fun <. u , t >. -> ( ( u = { x } /\ t = { x , y } ) -> x = y ) )
53		dfsn2 3389	. . . . . . . . . . 11 |- { x } = { x , x }
54		preq2 3448	. . . . . . . . . . 11 |- ( x = y -> { x , x } = { x , y } )
55	53, 54	syl5req 2085	. . . . . . . . . 10 |- ( x = y -> { x , y } = { x } )
56	55	eqeq2d 2051	. . . . . . . . 9 |- ( x = y -> ( t = { x , y } <-> t = { x } ) )
57		eqtr3 2059	. . . . . . . . . 10 |- ( ( u = { x } /\ t = { x } ) -> u = t )
58	57	expcom 109	. . . . . . . . 9 |- ( t = { x } -> ( u = { x } -> u = t ) )
59	56, 58	syl6bi 152	. . . . . . . 8 |- ( x = y -> ( t = { x , y } -> ( u = { x } -> u = t ) ) )
60	59	com13 74	. . . . . . 7 |- ( u = { x } -> ( t = { x , y } -> ( x = y -> u = t ) ) )
61	60	imp 115	. . . . . 6 |- ( ( u = { x } /\ t = { x , y } ) -> ( x = y -> u = t ) )
62	52, 61	sylcom 25	. . . . 5 |- ( Fun <. u , t >. -> ( ( u = { x } /\ t = { x , y } ) -> u = t ) )
63	62	exlimdvv 1777	. . . 4 |- ( Fun <. u , t >. -> ( E. x E. y ( u = { x } /\ t = { x , y } ) -> u = t ) )
64	13, 63	mpd 13	. . 3 |- ( Fun <. u , t >. -> u = t )
65	4, 8, 64	vtocl2g 2617	. 2 |- ( ( A e. V /\ B e. W ) -> ( Fun <. A , B >. -> A = B ) )
66	65	3impia 1101	1 |- ( ( A e. V /\ B e. W /\ Fun <. A , B >. ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557   {csn 3375   {cpr 3376   <.cop 3378   Rel wrel 4350   Fun wfun 4896

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-fun 4904

This theorem is referenced by: (None)