Theorem funopg 5324

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 3833	. . . . 5 |- ( u = A -> <. u , t >. = <. A , t >. )
2	1	funeqd 5312	. . . 4 |- ( u = A -> ( Fun <. u , t >. <-> Fun <. A , t >. ) )
3		eqeq1 2214	. . . 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 3834	. . . . 5 |- ( t = B -> <. A , t >. = <. A , B >. )
6	5	funeqd 5312	. . . 4 |- ( t = B -> ( Fun <. A , t >. <-> Fun <. A , B >. ) )
7		eqeq2 2217	. . . 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 5307	. . . . 5 |- ( Fun <. u , t >. -> Rel <. u , t >. )
10		vex 2779	. . . . . 6 |- u e. _V
11		vex 2779	. . . . . 6 |- t e. _V
12	10, 11	relop 4846	. . . . 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 4299	. . . . . . . 8 |- ( <. u , t >. = <. { x } , { x , y } >. <-> ( u = { x } /\ t = { x , y } ) )
15		vex 2779	. . . . . . . . . . . 12 |- x e. _V
16	15	opid 3851	. . . . . . . . . . 11 |- <. x , x >. = { { x } }
17	16	preq1i 3723	. . . . . . . . . 10 |- { <. x , x >. , { { x } , { x , y } } } = { { { x } } , { { x } , { x , y } } }
18		vex 2779	. . . . . . . . . . . 12 |- y e. _V
19	15, 18	dfop 3832	. . . . . . . . . . 11 |- <. x , y >. = { { x } , { x , y } }
20	19	preq2i 3724	. . . . . . . . . 10 |- { <. x , x >. , <. x , y >. } = { <. x , x >. , { { x } , { x , y } } }
21	15	snex 4245	. . . . . . . . . . 11 |- { x } e. _V
22		zfpair2 4270	. . . . . . . . . . 11 |- { x , y } e. _V
23	21, 22	dfop 3832	. . . . . . . . . 10 |- <. { x } , { x , y } >. = { { { x } } , { { x } , { x , y } } }
24	17, 20, 23	3eqtr4ri 2239	. . . . . . . . 9 |- <. { x } , { x , y } >. = { <. x , x >. , <. x , y >. }
25	24	eqeq2i 2218	. . . . . . . 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 5301	. . . . . . . . 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 4291	. . . . . . . . . . 11 |- <. x , x >. e. _V
30	29	prid1 3749	. . . . . . . . . 10 |- <. x , x >. e. { <. x , x >. , <. x , y >. }
31		eleq2 2271	. . . . . . . . . 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 4291	. . . . . . . . . . 11 |- <. x , y >. e. _V
34	33	prid2 3750	. . . . . . . . . 10 |- <. x , y >. e. { <. x , x >. , <. x , y >. }
35		eleq2 2271	. . . . . . . . . 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 3835	. . . . . . . . . . . . . 14 |- ( ( z = x /\ w = x ) -> <. z , w >. = <. x , x >. )
39	38	3adant3 1020	. . . . . . . . . . . . 13 |- ( ( z = x /\ w = x /\ v = y ) -> <. z , w >. = <. x , x >. )
40	39	eleq1d 2276	. . . . . . . . . . . 12 |- ( ( z = x /\ w = x /\ v = y ) -> ( <. z , w >. e. <. u , t >. <-> <. x , x >. e. <. u , t >. ) )
41		opeq12 3835	. . . . . . . . . . . . . 14 |- ( ( z = x /\ v = y ) -> <. z , v >. = <. x , y >. )
42	41	3adant2 1019	. . . . . . . . . . . . 13 |- ( ( z = x /\ w = x /\ v = y ) -> <. z , v >. = <. x , y >. )
43	42	eleq1d 2276	. . . . . . . . . . . 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 2220	. . . . . . . . . . . 12 |- ( ( w = x /\ v = y ) -> ( w = v <-> x = y ) )
46	45	3adant1 1018	. . . . . . . . . . 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 2873	. . . . . . . . 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 1350	. . . . . . . 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 3657	. . . . . . . . . . 11 |- { x } = { x , x }
53		preq2 3721	. . . . . . . . . . 11 |- ( x = y -> { x , x } = { x , y } )
54	52, 53	eqtr2id 2253	. . . . . . . . . 10 |- ( x = y -> { x , y } = { x } )
55	54	eqeq2d 2219	. . . . . . . . 9 |- ( x = y -> ( t = { x , y } <-> t = { x } ) )
56		eqtr3 2227	. . . . . . . . . 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 1922	. . . 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 2842	. 2 |- ( ( A e. V /\ B e. W ) -> ( Fun <. A , B >. -> A = B ) )
65	64	3impia 1203	1 |- ( ( A e. V /\ B e. W /\ Fun <. A , B >. ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   A.wal 1371   = wceq 1373   E.wex 1516   e. wcel 2178   _Vcvv 2776   {csn 3643   {cpr 3644   <.cop 3646   Rel wrel 4698   Fun wfun 5284

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-fun 5292

This theorem is referenced by:  funopsn  5785