Theorem exmidapne 7446

Description: Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.)

Assertion

Ref	Expression
exmidapne	|- (EXMID -> ( R TAp A <-> R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) )

Distinct variable group:   u, A, v

Allowed substitution hints:   R( v, u)

Proof of Theorem exmidapne

Dummy variables p x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 528	. . . . . . 7 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> R TAp A )
2		simpr 110	. . . . . . 7 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> p e. R )
3		dftap2 7437	. . . . . . . . . 10 |- ( R TAp A <-> ( R C_ ( A X. A ) /\ ( A. x e. A -. x R x /\ A. x e. A A. y e. A ( x R y -> y R x ) ) /\ ( A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) /\ A. x e. A A. y e. A ( -. x R y -> x = y ) ) ) )
4	3	biimpi 120	. . . . . . . . 9 |- ( R TAp A -> ( R C_ ( A X. A ) /\ ( A. x e. A -. x R x /\ A. x e. A A. y e. A ( x R y -> y R x ) ) /\ ( A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) /\ A. x e. A A. y e. A ( -. x R y -> x = y ) ) ) )
5	4	simp1d 1033	. . . . . . . 8 |- ( R TAp A -> R C_ ( A X. A ) )
6	5	sseld 3223	. . . . . . 7 |- ( R TAp A -> ( p e. R -> p e. ( A X. A ) ) )
7	1, 2, 6	sylc 62	. . . . . 6 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> p e. ( A X. A ) )
8		1st2nd2 6321	. . . . . 6 |- ( p e. ( A X. A ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
9	7, 8	syl 14	. . . . 5 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
10		xp1st 6311	. . . . . . . 8 |- ( p e. ( A X. A ) -> ( 1st ` p ) e. A )
11	7, 10	syl 14	. . . . . . 7 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> ( 1st ` p ) e. A )
12		xp2nd 6312	. . . . . . . 8 |- ( p e. ( A X. A ) -> ( 2nd ` p ) e. A )
13	7, 12	syl 14	. . . . . . 7 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> ( 2nd ` p ) e. A )
14	9, 2	eqeltrrd 2307	. . . . . . . . . 10 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> <. ( 1st ` p ) , ( 2nd ` p ) >. e. R )
15	14	adantr 276	. . . . . . . . 9 |- ( ( ( (EXMID /\ R TAp A ) /\ p e. R ) /\ ( 1st ` p ) = ( 2nd ` p ) ) -> <. ( 1st ` p ) , ( 2nd ` p ) >. e. R )
16		simpr 110	. . . . . . . . . . 11 |- ( ( ( (EXMID /\ R TAp A ) /\ p e. R ) /\ ( 1st ` p ) = ( 2nd ` p ) ) -> ( 1st ` p ) = ( 2nd ` p ) )
17	16	opeq2d 3864	. . . . . . . . . 10 |- ( ( ( (EXMID /\ R TAp A ) /\ p e. R ) /\ ( 1st ` p ) = ( 2nd ` p ) ) -> <. ( 1st ` p ) , ( 1st ` p ) >. = <. ( 1st ` p ) , ( 2nd ` p ) >. )
18		id 19	. . . . . . . . . . . . . 14 |- ( x = ( 1st ` p ) -> x = ( 1st ` p ) )
19	18, 18	breq12d 4096	. . . . . . . . . . . . 13 |- ( x = ( 1st ` p ) -> ( x R x <-> ( 1st ` p ) R ( 1st ` p ) ) )
20	19	notbid 671	. . . . . . . . . . . 12 |- ( x = ( 1st ` p ) -> ( -. x R x <-> -. ( 1st ` p ) R ( 1st ` p ) ) )
21	4	simp2d 1034	. . . . . . . . . . . . . 14 |- ( R TAp A -> ( A. x e. A -. x R x /\ A. x e. A A. y e. A ( x R y -> y R x ) ) )
22	21	simpld 112	. . . . . . . . . . . . 13 |- ( R TAp A -> A. x e. A -. x R x )
23	22	ad3antlr 493	. . . . . . . . . . . 12 |- ( ( ( (EXMID /\ R TAp A ) /\ p e. R ) /\ ( 1st ` p ) = ( 2nd ` p ) ) -> A. x e. A -. x R x )
24	11	adantr 276	. . . . . . . . . . . 12 |- ( ( ( (EXMID /\ R TAp A ) /\ p e. R ) /\ ( 1st ` p ) = ( 2nd ` p ) ) -> ( 1st ` p ) e. A )
25	20, 23, 24	rspcdva 2912	. . . . . . . . . . 11 |- ( ( ( (EXMID /\ R TAp A ) /\ p e. R ) /\ ( 1st ` p ) = ( 2nd ` p ) ) -> -. ( 1st ` p ) R ( 1st ` p ) )
26		df-br 4084	. . . . . . . . . . 11 |- ( ( 1st ` p ) R ( 1st ` p ) <-> <. ( 1st ` p ) , ( 1st ` p ) >. e. R )
27	25, 26	sylnib 680	. . . . . . . . . 10 |- ( ( ( (EXMID /\ R TAp A ) /\ p e. R ) /\ ( 1st ` p ) = ( 2nd ` p ) ) -> -. <. ( 1st ` p ) , ( 1st ` p ) >. e. R )
28	17, 27	eqneltrrd 2326	. . . . . . . . 9 |- ( ( ( (EXMID /\ R TAp A ) /\ p e. R ) /\ ( 1st ` p ) = ( 2nd ` p ) ) -> -. <. ( 1st ` p ) , ( 2nd ` p ) >. e. R )
29	15, 28	pm2.65da 665	. . . . . . . 8 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> -. ( 1st ` p ) = ( 2nd ` p ) )
30	29	neqned 2407	. . . . . . 7 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> ( 1st ` p ) =/= ( 2nd ` p ) )
31	11, 13, 30	jca31 309	. . . . . 6 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> ( ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) /\ ( 1st ` p ) =/= ( 2nd ` p ) ) )
32		eleq1 2292	. . . . . . . . . 10 |- ( u = ( 1st ` p ) -> ( u e. A <-> ( 1st ` p ) e. A ) )
33		eleq1 2292	. . . . . . . . . 10 |- ( v = ( 2nd ` p ) -> ( v e. A <-> ( 2nd ` p ) e. A ) )
34	32, 33	bi2anan9 608	. . . . . . . . 9 |- ( ( u = ( 1st ` p ) /\ v = ( 2nd ` p ) ) -> ( ( u e. A /\ v e. A ) <-> ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) ) )
35		simpl 109	. . . . . . . . . 10 |- ( ( u = ( 1st ` p ) /\ v = ( 2nd ` p ) ) -> u = ( 1st ` p ) )
36		simpr 110	. . . . . . . . . 10 |- ( ( u = ( 1st ` p ) /\ v = ( 2nd ` p ) ) -> v = ( 2nd ` p ) )
37	35, 36	neeq12d 2420	. . . . . . . . 9 |- ( ( u = ( 1st ` p ) /\ v = ( 2nd ` p ) ) -> ( u =/= v <-> ( 1st ` p ) =/= ( 2nd ` p ) ) )
38	34, 37	anbi12d 473	. . . . . . . 8 |- ( ( u = ( 1st ` p ) /\ v = ( 2nd ` p ) ) -> ( ( ( u e. A /\ v e. A ) /\ u =/= v ) <-> ( ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) /\ ( 1st ` p ) =/= ( 2nd ` p ) ) ) )
39	38	opelopabga 4351	. . . . . . 7 |- ( ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) -> ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } <-> ( ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) /\ ( 1st ` p ) =/= ( 2nd ` p ) ) ) )
40	11, 13, 39	syl2anc 411	. . . . . 6 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> ( <. ( 1st ` p ) , ( 2nd ` p ) >. e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } <-> ( ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) /\ ( 1st ` p ) =/= ( 2nd ` p ) ) ) )
41	31, 40	mpbird 167	. . . . 5 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> <. ( 1st ` p ) , ( 2nd ` p ) >. e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } )
42	9, 41	eqeltrd 2306	. . . 4 |- ( ( (EXMID /\ R TAp A ) /\ p e. R ) -> p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } )
43		relopab 4848	. . . . . . 7 |- Rel { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) }
44		1st2nd 6327	. . . . . . 7 |- ( ( Rel { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
45	43, 44	mpan 424	. . . . . 6 |- ( p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
46	45	adantl 277	. . . . 5 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
47		breq2 4087	. . . . . . . . . 10 |- ( y = ( 2nd ` p ) -> ( ( 1st ` p ) R y <-> ( 1st ` p ) R ( 2nd ` p ) ) )
48	47	notbid 671	. . . . . . . . 9 |- ( y = ( 2nd ` p ) -> ( -. ( 1st ` p ) R y <-> -. ( 1st ` p ) R ( 2nd ` p ) ) )
49		eqeq2 2239	. . . . . . . . 9 |- ( y = ( 2nd ` p ) -> ( ( 1st ` p ) = y <-> ( 1st ` p ) = ( 2nd ` p ) ) )
50	48, 49	imbi12d 234	. . . . . . . 8 |- ( y = ( 2nd ` p ) -> ( ( -. ( 1st ` p ) R y -> ( 1st ` p ) = y ) <-> ( -. ( 1st ` p ) R ( 2nd ` p ) -> ( 1st ` p ) = ( 2nd ` p ) ) ) )
51		breq1 4086	. . . . . . . . . . . 12 |- ( x = ( 1st ` p ) -> ( x R y <-> ( 1st ` p ) R y ) )
52	51	notbid 671	. . . . . . . . . . 11 |- ( x = ( 1st ` p ) -> ( -. x R y <-> -. ( 1st ` p ) R y ) )
53		eqeq1 2236	. . . . . . . . . . 11 |- ( x = ( 1st ` p ) -> ( x = y <-> ( 1st ` p ) = y ) )
54	52, 53	imbi12d 234	. . . . . . . . . 10 |- ( x = ( 1st ` p ) -> ( ( -. x R y -> x = y ) <-> ( -. ( 1st ` p ) R y -> ( 1st ` p ) = y ) ) )
55	54	ralbidv 2530	. . . . . . . . 9 |- ( x = ( 1st ` p ) -> ( A. y e. A ( -. x R y -> x = y ) <-> A. y e. A ( -. ( 1st ` p ) R y -> ( 1st ` p ) = y ) ) )
56	4	simp3d 1035	. . . . . . . . . . 11 |- ( R TAp A -> ( A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ y R z ) ) /\ A. x e. A A. y e. A ( -. x R y -> x = y ) ) )
57	56	simprd 114	. . . . . . . . . 10 |- ( R TAp A -> A. x e. A A. y e. A ( -. x R y -> x = y ) )
58	57	ad2antlr 489	. . . . . . . . 9 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> A. x e. A A. y e. A ( -. x R y -> x = y ) )
59	32	anbi1d 465	. . . . . . . . . . . . . 14 |- ( u = ( 1st ` p ) -> ( ( u e. A /\ v e. A ) <-> ( ( 1st ` p ) e. A /\ v e. A ) ) )
60		neeq1 2413	. . . . . . . . . . . . . 14 |- ( u = ( 1st ` p ) -> ( u =/= v <-> ( 1st ` p ) =/= v ) )
61	59, 60	anbi12d 473	. . . . . . . . . . . . 13 |- ( u = ( 1st ` p ) -> ( ( ( u e. A /\ v e. A ) /\ u =/= v ) <-> ( ( ( 1st ` p ) e. A /\ v e. A ) /\ ( 1st ` p ) =/= v ) ) )
62	33	anbi2d 464	. . . . . . . . . . . . . 14 |- ( v = ( 2nd ` p ) -> ( ( ( 1st ` p ) e. A /\ v e. A ) <-> ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) ) )
63		neeq2 2414	. . . . . . . . . . . . . 14 |- ( v = ( 2nd ` p ) -> ( ( 1st ` p ) =/= v <-> ( 1st ` p ) =/= ( 2nd ` p ) ) )
64	62, 63	anbi12d 473	. . . . . . . . . . . . 13 |- ( v = ( 2nd ` p ) -> ( ( ( ( 1st ` p ) e. A /\ v e. A ) /\ ( 1st ` p ) =/= v ) <-> ( ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) /\ ( 1st ` p ) =/= ( 2nd ` p ) ) ) )
65	61, 64	elopabi 6341	. . . . . . . . . . . 12 |- ( p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } -> ( ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) /\ ( 1st ` p ) =/= ( 2nd ` p ) ) )
66	65	adantl 277	. . . . . . . . . . 11 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> ( ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) /\ ( 1st ` p ) =/= ( 2nd ` p ) ) )
67	66	simpld 112	. . . . . . . . . 10 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> ( ( 1st ` p ) e. A /\ ( 2nd ` p ) e. A ) )
68	67	simpld 112	. . . . . . . . 9 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> ( 1st ` p ) e. A )
69	55, 58, 68	rspcdva 2912	. . . . . . . 8 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> A. y e. A ( -. ( 1st ` p ) R y -> ( 1st ` p ) = y ) )
70	67	simprd 114	. . . . . . . 8 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> ( 2nd ` p ) e. A )
71	50, 69, 70	rspcdva 2912	. . . . . . 7 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> ( -. ( 1st ` p ) R ( 2nd ` p ) -> ( 1st ` p ) = ( 2nd ` p ) ) )
72	66	simprd 114	. . . . . . . 8 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> ( 1st ` p ) =/= ( 2nd ` p ) )
73	72	neneqd 2421	. . . . . . 7 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> -. ( 1st ` p ) = ( 2nd ` p ) )
74		exmidexmid 4280	. . . . . . . . 9 |- (EXMID -> DECID ( 1st ` p ) R ( 2nd ` p ) )
75		con1dc 861	. . . . . . . . 9 |- (DECID ( 1st ` p ) R ( 2nd ` p ) -> ( ( -. ( 1st ` p ) R ( 2nd ` p ) -> ( 1st ` p ) = ( 2nd ` p ) ) -> ( -. ( 1st ` p ) = ( 2nd ` p ) -> ( 1st ` p ) R ( 2nd ` p ) ) ) )
76	74, 75	syl 14	. . . . . . . 8 |- (EXMID -> ( ( -. ( 1st ` p ) R ( 2nd ` p ) -> ( 1st ` p ) = ( 2nd ` p ) ) -> ( -. ( 1st ` p ) = ( 2nd ` p ) -> ( 1st ` p ) R ( 2nd ` p ) ) ) )
77	76	ad2antrr 488	. . . . . . 7 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> ( ( -. ( 1st ` p ) R ( 2nd ` p ) -> ( 1st ` p ) = ( 2nd ` p ) ) -> ( -. ( 1st ` p ) = ( 2nd ` p ) -> ( 1st ` p ) R ( 2nd ` p ) ) ) )
78	71, 73, 77	mp2d 47	. . . . . 6 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> ( 1st ` p ) R ( 2nd ` p ) )
79		df-br 4084	. . . . . 6 |- ( ( 1st ` p ) R ( 2nd ` p ) <-> <. ( 1st ` p ) , ( 2nd ` p ) >. e. R )
80	78, 79	sylib 122	. . . . 5 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> <. ( 1st ` p ) , ( 2nd ` p ) >. e. R )
81	46, 80	eqeltrd 2306	. . . 4 |- ( ( (EXMID /\ R TAp A ) /\ p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> p e. R )
82	42, 81	impbida 598	. . 3 |- ( (EXMID /\ R TAp A ) -> ( p e. R <-> p e. { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) )
83	82	eqrdv 2227	. 2 |- ( (EXMID /\ R TAp A ) -> R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } )
84		exmidexmid 4280	. . . . . . 7 |- (EXMID -> DECID x = y )
85	84	ralrimivw 2604	. . . . . 6 |- (EXMID -> A. y e. A DECID x = y )
86	85	ralrimivw 2604	. . . . 5 |- (EXMID -> A. x e. A A. y e. A DECID x = y )
87		netap 7440	. . . . 5 |- ( A. x e. A A. y e. A DECID x = y -> { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } TAp A )
88	86, 87	syl 14	. . . 4 |- (EXMID -> { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } TAp A )
89	88	adantr 276	. . 3 |- ( (EXMID /\ R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } TAp A )
90		tapeq1 7438	. . . 4 |- ( R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } -> ( R TAp A <-> { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } TAp A ) )
91	90	adantl 277	. . 3 |- ( (EXMID /\ R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> ( R TAp A <-> { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } TAp A ) )
92	89, 91	mpbird 167	. 2 |- ( (EXMID /\ R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) -> R TAp A )
93	83, 92	impbida 598	1 |- (EXMID -> ( R TAp A <-> R = { <. u , v >. | ( ( u e. A /\ v e. A ) /\ u =/= v ) } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   C_ wss 3197   <.cop 3669   class class class wbr 4083   {copab 4144  EXMIDwem 4278   X. cxp 4717   Rel wrel 4724   ` cfv 5318   1stc1st 6284   2ndc2nd 6285   TAp wtap 7435

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-exmid 4279  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-1st 6286  df-2nd 6287  df-pap 7434  df-tap 7436

This theorem is referenced by:  exmidmotap  7447