Theorem funopg 5386

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 3883	. . . . 5 |- ( u = A -> <. u , t >. = <. A , t >. )
2	1	funeqd 5374	. . . 4 |- ( u = A -> ( Fun <. u , t >. <-> Fun <. A , t >. ) )
3		eqeq1 2239	. . . 4 |- ( u = A -> ( u = t <-> A = t ) )
4	2, 3	imbi12d 234	. . 3 |- ( u = A -> ( ( Fun <. u , t >. -> u = t ) <-> ( Fun <. A , t >. -> A = t ) ) )
5		opeq2 3884	. . . . 5 |- ( t = B -> <. A , t >. = <. A , B >. )
6	5	funeqd 5374	. . . 4 |- ( t = B -> ( Fun <. A , t >. <-> Fun <. A , B >. ) )
7		eqeq2 2242	. . . 4 |- ( t = B -> ( A = t <-> A = B ) )
8	6, 7	imbi12d 234	. . 3 |- ( t = B -> ( ( Fun <. A , t >. -> A = t ) <-> ( Fun <. A , B >. -> A = B ) ) )
9		funrel 5369	. . . . 5 |- ( Fun <. u , t >. -> Rel <. u , t >. )
10		vex 2816	. . . . . 6 |- u e. _V
11		vex 2816	. . . . . 6 |- t e. _V
12	10, 11	relop 4905	. . . . 5 |- ( Rel <. u , t >. <-> E. x E. y ( u = { x } /\ t = { x , y } ) )
13	9, 12	sylib 122	. . . 4 |- ( Fun <. u , t >. -> E. x E. y ( u = { x } /\ t = { x , y } ) )
14	10, 11	opth 4353	. . . . . . . 8 |- ( <. u , t >. = <. { x } , { x , y } >. <-> ( u = { x } /\ t = { x , y } ) )
15		vex 2816	. . . . . . . . . . . 12 |- x e. _V
16	15	opid 3901	. . . . . . . . . . 11 |- <. x , x >. = { { x } }
17	16	preq1i 3771	. . . . . . . . . 10 |- { <. x , x >. , { { x } , { x , y } } } = { { { x } } , { { x } , { x , y } } }
18		vex 2816	. . . . . . . . . . . 12 |- y e. _V
19	15, 18	dfop 3882	. . . . . . . . . . 11 |- <. x , y >. = { { x } , { x , y } }
20	19	preq2i 3772	. . . . . . . . . 10 |- { <. x , x >. , <. x , y >. } = { <. x , x >. , { { x } , { x , y } } }
21	15	snex 4298	. . . . . . . . . . 11 |- { x } e. _V
22		zfpair2 4323	. . . . . . . . . . 11 |- { x , y } e. _V
23	21, 22	dfop 3882	. . . . . . . . . 10 |- <. { x } , { x , y } >. = { { { x } } , { { x } , { x , y } } }
24	17, 20, 23	3eqtr4ri 2264	. . . . . . . . 9 |- <. { x } , { x , y } >. = { <. x , x >. , <. x , y >. }
25	24	eqeq2i 2243	. . . . . . . 8 |- ( <. u , t >. = <. { x } , { x , y } >. <-> <. u , t >. = { <. x , x >. , <. x , y >. } )
26	14, 25	bitr3i 186	. . . . . . 7 |- ( ( u = { x } /\ t = { x , y } ) <-> <. u , t >. = { <. x , x >. , <. x , y >. } )
27		dffun4 5363	. . . . . . . . 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 275	. . . . . . . 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 4345	. . . . . . . . . . 11 |- <. x , x >. e. _V
30	29	prid1 3797	. . . . . . . . . 10 |- <. x , x >. e. { <. x , x >. , <. x , y >. }
31		eleq2 2296	. . . . . . . . . 10 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> ( <. x , x >. e. <. u , t >. <-> <. x , x >. e. { <. x , x >. , <. x , y >. } ) )
32	30, 31	mpbiri 168	. . . . . . . . 9 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> <. x , x >. e. <. u , t >. )
33	15, 18	opex 4345	. . . . . . . . . . 11 |- <. x , y >. e. _V
34	33	prid2 3798	. . . . . . . . . 10 |- <. x , y >. e. { <. x , x >. , <. x , y >. }
35		eleq2 2296	. . . . . . . . . 10 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> ( <. x , y >. e. <. u , t >. <-> <. x , y >. e. { <. x , x >. , <. x , y >. } ) )
36	34, 35	mpbiri 168	. . . . . . . . 9 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> <. x , y >. e. <. u , t >. )
37	32, 36	jca 306	. . . . . . . 8 |- ( <. u , t >. = { <. x , x >. , <. x , y >. } -> ( <. x , x >. e. <. u , t >. /\ <. x , y >. e. <. u , t >. ) )
38		opeq12 3885	. . . . . . . . . . . . . 14 |- ( ( z = x /\ w = x ) -> <. z , w >. = <. x , x >. )
39	38	3adant3 1044	. . . . . . . . . . . . 13 |- ( ( z = x /\ w = x /\ v = y ) -> <. z , w >. = <. x , x >. )
40	39	eleq1d 2301	. . . . . . . . . . . 12 |- ( ( z = x /\ w = x /\ v = y ) -> ( <. z , w >. e. <. u , t >. <-> <. x , x >. e. <. u , t >. ) )
41		opeq12 3885	. . . . . . . . . . . . . 14 |- ( ( z = x /\ v = y ) -> <. z , v >. = <. x , y >. )
42	41	3adant2 1043	. . . . . . . . . . . . 13 |- ( ( z = x /\ w = x /\ v = y ) -> <. z , v >. = <. x , y >. )
43	42	eleq1d 2301	. . . . . . . . . . . 12 |- ( ( z = x /\ w = x /\ v = y ) -> ( <. z , v >. e. <. u , t >. <-> <. x , y >. e. <. u , t >. ) )
44	40, 43	anbi12d 473	. . . . . . . . . . 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 2245	. . . . . . . . . . . 12 |- ( ( w = x /\ v = y ) -> ( w = v <-> x = y ) )
46	45	3adant1 1042	. . . . . . . . . . 11 |- ( ( z = x /\ w = x /\ v = y ) -> ( w = v <-> x = y ) )
47	44, 46	imbi12d 234	. . . . . . . . . 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 2910	. . . . . . . . 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 1374	. . . . . . . 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	biimtrid 152	. . . . . 6 |- ( Fun <. u , t >. -> ( ( u = { x } /\ t = { x , y } ) -> x = y ) )
52		dfsn2 3703	. . . . . . . . . . 11 |- { x } = { x , x }
53		preq2 3769	. . . . . . . . . . 11 |- ( x = y -> { x , x } = { x , y } )
54	52, 53	eqtr2id 2278	. . . . . . . . . 10 |- ( x = y -> { x , y } = { x } )
55	54	eqeq2d 2244	. . . . . . . . 9 |- ( x = y -> ( t = { x , y } <-> t = { x } ) )
56		eqtr3 2252	. . . . . . . . . 10 |- ( ( u = { x } /\ t = { x } ) -> u = t )
57	56	expcom 116	. . . . . . . . 9 |- ( t = { x } -> ( u = { x } -> u = t ) )
58	55, 57	biimtrdi 163	. . . . . . . 8 |- ( x = y -> ( t = { x , y } -> ( u = { x } -> u = t ) ) )
59	58	com13 80	. . . . . . 7 |- ( u = { x } -> ( t = { x , y } -> ( x = y -> u = t ) ) )
60	59	imp 124	. . . . . 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 1947	. . . 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 2879	. 2 |- ( ( A e. V /\ B e. W ) -> ( Fun <. A , B >. -> A = B ) )
65	64	3impia 1227	1 |- ( ( A e. V /\ B e. W /\ Fun <. A , B >. ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   A.wal 1396   = wceq 1398   E.wex 1541   e. wcel 2203   _Vcvv 2813   {csn 3689   {cpr 3690   <.cop 3692   Rel wrel 4754   Fun wfun 5346

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-fun 5354

This theorem is referenced by:  funopsn  5860