Theorem funopg 4954

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 3570	. . . . 5 |- ( u = A -> <. u , t >. = <. A , t >. )
2	1	funeqd 4943	. . . 4 |- ( u = A -> ( Fun <. u , t >. <-> Fun <. A , t >. ) )
3		eqeq1 2087	. . . 4 |- ( u = A -> ( u = t <-> A = t ) )
4	2, 3	imbi12d 232	. . 3 |- ( u = A -> ( ( Fun <. u , t >. -> u = t ) <-> ( Fun <. A , t >. -> A = t ) ) )
5		opeq2 3571	. . . . 5 |- ( t = B -> <. A , t >. = <. A , B >. )
6	5	funeqd 4943	. . . 4 |- ( t = B -> ( Fun <. A , t >. <-> Fun <. A , B >. ) )
7		eqeq2 2090	. . . 4 |- ( t = B -> ( A = t <-> A = B ) )
8	6, 7	imbi12d 232	. . 3 |- ( t = B -> ( ( Fun <. A , t >. -> A = t ) <-> ( Fun <. A , B >. -> A = B ) ) )
9		funrel 4939	. . . . 5 |- ( Fun <. u , t >. -> Rel <. u , t >. )
10		vex 2604	. . . . . 6 |- u e. _V
11		vex 2604	. . . . . 6 |- t e. _V
12	10, 11	relop 4504	. . . . 5 |- ( Rel <. u , t >. <-> E. x E. y ( u = { x } /\ t = { x , y } ) )
13	9, 12	sylib 120	. . . 4 |- ( Fun <. u , t >. -> E. x E. y ( u = { x } /\ t = { x , y } ) )
14	10, 11	opth 3992	. . . . . . . 8 |- ( <. u , t >. = <. { x } , { x , y } >. <-> ( u = { x } /\ t = { x , y } ) )
15		vex 2604	. . . . . . . . . . . 12 |- x e. _V
16	15	opid 3588	. . . . . . . . . . 11 |- <. x , x >. = { { x } }
17	16	preq1i 3472	. . . . . . . . . 10 |- { <. x , x >. , { { x } , { x , y } } } = { { { x } } , { { x } , { x , y } } }
18		vex 2604	. . . . . . . . . . . 12 |- y e. _V
19	15, 18	dfop 3569	. . . . . . . . . . 11 |- <. x , y >. = { { x } , { x , y } }
20	19	preq2i 3473	. . . . . . . . . 10 |- { <. x , x >. , <. x , y >. } = { <. x , x >. , { { x } , { x , y } } }
21	15	snex 3957	. . . . . . . . . . 11 |- { x } e. _V
22		zfpair2 3965	. . . . . . . . . . 11 |- { x , y } e. _V
23	21, 22	dfop 3569	. . . . . . . . . 10 |- <. { x } , { x , y } >. = { { { x } } , { { x } , { x , y } } }
24	17, 20, 23	3eqtr4ri 2112	. . . . . . . . 9 |- <. { x } , { x , y } >. = { <. x , x >. , <. x , y >. }
25	24	eqeq2i 2091	. . . . . . . 8 |- ( <. u , t >. = <. { x } , { x , y } >. <-> <. u , t >. = { <. x , x >. , <. x , y >. } )
26	14, 25	bitr3i 184	. . . . . . 7 |- ( ( u = { x } /\ t = { x , y } ) <-> <. u , t >. = { <. x , x >. , <. x , y >. } )
27		dffun4 4933	. . . . . . . . 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 ) ) )
28	27	simprbi 269	. . . . . . . 8 |- ( Fun <. u , t >. -> A. z A. w A. v ( ( <. z , w >. e. <. u , t >. /\ <. z , v >. e. <. u , t >. ) -> w = v ) )
29	15, 15	opex 3984	. . . . . . . . . . 11 |- <. x , x >. e. _V
30	29	prid1 3498	. . . . . . . . . 10 |- <. x , x >. e. { <. x , x >. , <. x , y >. }
31		eleq2 2142	. . . . . . . . . 10 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> ( <. x , x >. e. <. u , t >. <-> <. x , x >. e. { <. x , x >. , <. x , y >. } ) )
32	30, 31	mpbiri 166	. . . . . . . . 9 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> <. x , x >. e. <. u , t >. )
33	15, 18	opex 3984	. . . . . . . . . . 11 |- <. x , y >. e. _V
34	33	prid2 3499	. . . . . . . . . 10 |- <. x , y >. e. { <. x , x >. , <. x , y >. }
35		eleq2 2142	. . . . . . . . . 10 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> ( <. x , y >. e. <. u , t >. <-> <. x , y >. e. { <. x , x >. , <. x , y >. } ) )
36	34, 35	mpbiri 166	. . . . . . . . 9 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> <. x , y >. e. <. u , t >. )
37	32, 36	jca 300	. . . . . . . 8 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> ( <. x , x >. e. <. u , t >. /\ <. x , y >. e. <. u , t >. ) )
38		opeq12 3572	. . . . . . . . . . . . . 14 |- ( ( z = x /\ w = x ) -> <. z , w >. = <. x , x >. )
39	38	3adant3 958	. . . . . . . . . . . . 13 |- ( ( z = x /\ w = x /\ v = y ) -> <. z , w >. = <. x , x >. )
40	39	eleq1d 2147	. . . . . . . . . . . 12 |- ( ( z = x /\ w = x /\ v = y ) -> ( <. z , w >. e. <. u , t >. <-> <. x , x >. e. <. u , t >. ) )
41		opeq12 3572	. . . . . . . . . . . . . 14 |- ( ( z = x /\ v = y ) -> <. z , v >. = <. x , y >. )
42	41	3adant2 957	. . . . . . . . . . . . 13 |- ( ( z = x /\ w = x /\ v = y ) -> <. z , v >. = <. x , y >. )
43	42	eleq1d 2147	. . . . . . . . . . . 12 |- ( ( z = x /\ w = x /\ v = y ) -> ( <. z , v >. e. <. u , t >. <-> <. x , y >. e. <. u , t >. ) )
44	40, 43	anbi12d 456	. . . . . . . . . . 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 >. ) ) )
45		eqeq12 2093	. . . . . . . . . . . 12 |- ( ( w = x /\ v = y ) -> ( w = v <-> x = y ) )
46	45	3adant1 956	. . . . . . . . . . 11 |- ( ( z = x /\ w = x /\ v = y ) -> ( w = v <-> x = y ) )
47	44, 46	imbi12d 232	. . . . . . . . . 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 ) ) )
48	47	spc3gv 2690	. . . . . . . . 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 ) ) )
49	15, 15, 18, 48	mp3an 1268	. . . . . . . 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 ) )
50	28, 37, 49	syl2im 38	. . . . . . 7 |- ( Fun <. u , t >. -> ( <. u , t >. = { <. x , x >. , <. x , y >. } -> x = y ) )
51	26, 50	syl5bi 150	. . . . . 6 |- ( Fun <. u , t >. -> ( ( u = { x } /\ t = { x , y } ) -> x = y ) )
52		dfsn2 3412	. . . . . . . . . . 11 |- { x } = { x , x }
53		preq2 3470	. . . . . . . . . . 11 |- ( x = y -> { x , x } = { x , y } )
54	52, 53	syl5req 2126	. . . . . . . . . 10 |- ( x = y -> { x , y } = { x } )
55	54	eqeq2d 2092	. . . . . . . . 9 |- ( x = y -> ( t = { x , y } <-> t = { x } ) )
56		eqtr3 2100	. . . . . . . . . 10 |- ( ( u = { x } /\ t = { x } ) -> u = t )
57	56	expcom 114	. . . . . . . . 9 |- ( t = { x } -> ( u = { x } -> u = t ) )
58	55, 57	syl6bi 161	. . . . . . . 8 |- ( x = y -> ( t = { x , y } -> ( u = { x } -> u = t ) ) )
59	58	com13 79	. . . . . . 7 |- ( u = { x } -> ( t = { x , y } -> ( x = y -> u = t ) ) )
60	59	imp 122	. . . . . 6 |- ( ( u = { x } /\ t = { x , y } ) -> ( x = y -> u = t ) )
61	51, 60	sylcom 28	. . . . 5 |- ( Fun <. u , t >. -> ( ( u = { x } /\ t = { x , y } ) -> u = t ) )
62	61	exlimdvv 1818	. . . 4 |- ( Fun <. u , t >. -> ( E. x E. y ( u = { x } /\ t = { x , y } ) -> u = t ) )
63	13, 62	mpd 13	. . 3 |- ( Fun <. u , t >. -> u = t )
64	4, 8, 63	vtocl2g 2662	. 2 |- ( ( A e. V /\ B e. W ) -> ( Fun <. A , B >. -> A = B ) )
65	64	3impia 1135	1 |- ( ( A e. V /\ B e. W /\ Fun <. A , B >. ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   {csn 3398   {cpr 3399   <.cop 3401   Rel wrel 4368   Fun wfun 4916

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-fun 4924

This theorem is referenced by: (None)