Theorem pstmfval 26259

Description: Function value of the metric induced by a pseudometric D (Contributed by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

Ref	Expression
pstmval.1	|- .~ = (~Met ` D )

Assertion

Ref	Expression
pstmfval	|- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( [ A ] .~ (pstoMet ` D ) [ B ] .~ ) = ( A D B ) )

Proof of Theorem pstmfval

Dummy variables a b x y z e f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pstmval.1	. . . . 5 |- .~ = (~Met ` D )
2	1	pstmval 26258	. . . 4 |- ( D e. (PsMet ` X ) -> (pstoMet ` D ) = ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } ) )
3	2	3ad2ant1 1004	. . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> (pstoMet ` D ) = ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } ) )
4	3	oveqd 6107	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( [ A ] .~ (pstoMet ` D ) [ B ] .~ ) = ( [ A ] .~ ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } ) [ B ] .~ ) )
5		fvex 5698	. . . . . 6 |- (~Met ` D ) e. _V
6	1, 5	eqeltri 2511	. . . . 5 |- .~ e. _V
7	6	ecelqsi 7152	. . . 4 |- ( A e. X -> [ A ] .~ e. ( X /. .~ ) )
8	7	3ad2ant2 1005	. . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> [ A ] .~ e. ( X /. .~ ) )
9	6	ecelqsi 7152	. . . 4 |- ( B e. X -> [ B ] .~ e. ( X /. .~ ) )
10	9	3ad2ant3 1006	. . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> [ B ] .~ e. ( X /. .~ ) )
11		rexeq 2916	. . . . . 6 |- ( x = [ A ] .~ -> ( E. a e. x E. b e. y z = ( a D b ) <-> E. a e. [ A ] .~ E. b e. y z = ( a D b ) ) )
12	11	abbidv 2555	. . . . 5 |- ( x = [ A ] .~ -> { z | E. a e. x E. b e. y z = ( a D b ) } = { z | E. a e. [ A ] .~ E. b e. y z = ( a D b ) } )
13	12	unieqd 4098	. . . 4 |- ( x = [ A ] .~ -> U. { z | E. a e. x E. b e. y z = ( a D b ) } = U. { z | E. a e. [ A ] .~ E. b e. y z = ( a D b ) } )
14		rexeq 2916	. . . . . . 7 |- ( y = [ B ] .~ -> ( E. b e. y z = ( a D b ) <-> E. b e. [ B ] .~ z = ( a D b ) ) )
15	14	rexbidv 2734	. . . . . 6 |- ( y = [ B ] .~ -> ( E. a e. [ A ] .~ E. b e. y z = ( a D b ) <-> E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) ) )
16	15	abbidv 2555	. . . . 5 |- ( y = [ B ] .~ -> { z | E. a e. [ A ] .~ E. b e. y z = ( a D b ) } = { z | E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) } )
17	16	unieqd 4098	. . . 4 |- ( y = [ B ] .~ -> U. { z | E. a e. [ A ] .~ E. b e. y z = ( a D b ) } = U. { z | E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) } )
18		eqid 2441	. . . 4 |- ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } ) = ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } )
19		ecexg 7101	. . . . . . 7 |- ( .~ e. _V -> [ A ] .~ e. _V )
20	6, 19	ax-mp 5	. . . . . 6 |- [ A ] .~ e. _V
21		ecexg 7101	. . . . . . 7 |- ( .~ e. _V -> [ B ] .~ e. _V )
22	6, 21	ax-mp 5	. . . . . 6 |- [ B ] .~ e. _V
23	20, 22	ab2rexex 6567	. . . . 5 |- { z | E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) } e. _V
24	23	uniex 6375	. . . 4 |- U. { z | E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) } e. _V
25	13, 17, 18, 24	ovmpt2 6225	. . 3 |- ( ( [ A ] .~ e. ( X /. .~ ) /\ [ B ] .~ e. ( X /. .~ ) ) -> ( [ A ] .~ ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } ) [ B ] .~ ) = U. { z | E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) } )
26	8, 10, 25	syl2anc 656	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( [ A ] .~ ( x e. ( X /. .~ ) , y e. ( X /. .~ ) |-> U. { z | E. a e. x E. b e. y z = ( a D b ) } ) [ B ] .~ ) = U. { z | E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) } )
27		simpr3 991	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> z = ( e D f ) )
28		simpl1 986	. . . . . . . . . . . . 13 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> D e. (PsMet ` X ) )
29		simpr1 989	. . . . . . . . . . . . . . 15 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> e e. [ A ] .~ )
30		metidss 26254	. . . . . . . . . . . . . . . . . . . . 21 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( X X. X ) )
31	1, 30	syl5eqss 3397	. . . . . . . . . . . . . . . . . . . 20 |- ( D e. (PsMet ` X ) -> .~ C_ ( X X. X ) )
32		xpss 4942	. . . . . . . . . . . . . . . . . . . 20 |- ( X X. X ) C_ ( _V X. _V )
33	31, 32	syl6ss 3365	. . . . . . . . . . . . . . . . . . 19 |- ( D e. (PsMet ` X ) -> .~ C_ ( _V X. _V ) )
34		df-rel 4843	. . . . . . . . . . . . . . . . . . 19 |- ( Rel .~ <-> .~ C_ ( _V X. _V ) )
35	33, 34	sylibr 212	. . . . . . . . . . . . . . . . . 18 |- ( D e. (PsMet ` X ) -> Rel .~ )
36	35	3ad2ant1 1004	. . . . . . . . . . . . . . . . 17 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> Rel .~ )
37	36	adantr 462	. . . . . . . . . . . . . . . 16 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> Rel .~ )
38		relelec 7137	. . . . . . . . . . . . . . . 16 |- ( Rel .~ -> ( e e. [ A ] .~ <-> A .~ e ) )
39	37, 38	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> ( e e. [ A ] .~ <-> A .~ e ) )
40	29, 39	mpbid 210	. . . . . . . . . . . . . 14 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> A .~ e )
41	1	breqi 4295	. . . . . . . . . . . . . 14 |- ( A .~ e <-> A (~Met ` D ) e )
42	40, 41	sylib 196	. . . . . . . . . . . . 13 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> A (~Met ` D ) e )
43		simpr2 990	. . . . . . . . . . . . . . 15 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> f e. [ B ] .~ )
44		relelec 7137	. . . . . . . . . . . . . . . 16 |- ( Rel .~ -> ( f e. [ B ] .~ <-> B .~ f ) )
45	37, 44	syl 16	. . . . . . . . . . . . . . 15 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> ( f e. [ B ] .~ <-> B .~ f ) )
46	43, 45	mpbid 210	. . . . . . . . . . . . . 14 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> B .~ f )
47	1	breqi 4295	. . . . . . . . . . . . . 14 |- ( B .~ f <-> B (~Met ` D ) f )
48	46, 47	sylib 196	. . . . . . . . . . . . 13 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> B (~Met ` D ) f )
49		metideq 26256	. . . . . . . . . . . . 13 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) e /\ B (~Met ` D ) f ) ) -> ( A D B ) = ( e D f ) )
50	28, 42, 48, 49	syl12anc 1211	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> ( A D B ) = ( e D f ) )
51	27, 50	eqtr4d 2476	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> z = ( A D B ) )
52	51	adantlr 709	. . . . . . . . . 10 |- ( ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) ) /\ ( e e. [ A ] .~ /\ f e. [ B ] .~ /\ z = ( e D f ) ) ) -> z = ( A D B ) )
53	52	3anassrs 1204	. . . . . . . . 9 |- ( ( ( ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) ) /\ e e. [ A ] .~ ) /\ f e. [ B ] .~ ) /\ z = ( e D f ) ) -> z = ( A D B ) )
54		oveq1 6097	. . . . . . . . . . . . 13 |- ( a = e -> ( a D b ) = ( e D b ) )
55	54	eqeq2d 2452	. . . . . . . . . . . 12 |- ( a = e -> ( z = ( a D b ) <-> z = ( e D b ) ) )
56		oveq2 6098	. . . . . . . . . . . . 13 |- ( b = f -> ( e D b ) = ( e D f ) )
57	56	eqeq2d 2452	. . . . . . . . . . . 12 |- ( b = f -> ( z = ( e D b ) <-> z = ( e D f ) ) )
58	55, 57	cbvrex2v 2954	. . . . . . . . . . 11 |- ( E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) <-> E. e e. [ A ] .~ E. f e. [ B ] .~ z = ( e D f ) )
59	58	biimpi 194	. . . . . . . . . 10 |- ( E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) -> E. e e. [ A ] .~ E. f e. [ B ] .~ z = ( e D f ) )
60	59	adantl 463	. . . . . . . . 9 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) ) -> E. e e. [ A ] .~ E. f e. [ B ] .~ z = ( e D f ) )
61	53, 60	r19.29_2a 2862	. . . . . . . 8 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) ) -> z = ( A D B ) )
62	61	ex 434	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) -> z = ( A D B ) ) )
63		simpl1 986	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> D e. (PsMet ` X ) )
64		simpl2 987	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> A e. X )
65		psmet0 19843	. . . . . . . . . . 11 |- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )
66	63, 64, 65	syl2anc 656	. . . . . . . . . 10 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> ( A D A ) = 0 )
67		relelec 7137	. . . . . . . . . . . 12 |- ( Rel .~ -> ( A e. [ A ] .~ <-> A .~ A ) )
68	63, 35, 67	3syl 20	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> ( A e. [ A ] .~ <-> A .~ A ) )
69	1	a1i 11	. . . . . . . . . . . 12 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> .~ = (~Met ` D ) )
70	69	breqd 4300	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> ( A .~ A <-> A (~Met ` D ) A ) )
71		metidv 26255	. . . . . . . . . . . 12 |- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ A e. X ) ) -> ( A (~Met ` D ) A <-> ( A D A ) = 0 ) )
72	63, 64, 64, 71	syl12anc 1211	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> ( A (~Met ` D ) A <-> ( A D A ) = 0 ) )
73	68, 70, 72	3bitrd 279	. . . . . . . . . 10 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> ( A e. [ A ] .~ <-> ( A D A ) = 0 ) )
74	66, 73	mpbird 232	. . . . . . . . 9 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> A e. [ A ] .~ )
75		simpl3 988	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> B e. X )
76		psmet0 19843	. . . . . . . . . . 11 |- ( ( D e. (PsMet ` X ) /\ B e. X ) -> ( B D B ) = 0 )
77	63, 75, 76	syl2anc 656	. . . . . . . . . 10 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> ( B D B ) = 0 )
78		relelec 7137	. . . . . . . . . . . 12 |- ( Rel .~ -> ( B e. [ B ] .~ <-> B .~ B ) )
79	63, 35, 78	3syl 20	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> ( B e. [ B ] .~ <-> B .~ B ) )
80	69	breqd 4300	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> ( B .~ B <-> B (~Met ` D ) B ) )
81		metidv 26255	. . . . . . . . . . . 12 |- ( ( D e. (PsMet ` X ) /\ ( B e. X /\ B e. X ) ) -> ( B (~Met ` D ) B <-> ( B D B ) = 0 ) )
82	63, 75, 75, 81	syl12anc 1211	. . . . . . . . . . 11 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> ( B (~Met ` D ) B <-> ( B D B ) = 0 ) )
83	79, 80, 82	3bitrd 279	. . . . . . . . . 10 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> ( B e. [ B ] .~ <-> ( B D B ) = 0 ) )
84	77, 83	mpbird 232	. . . . . . . . 9 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> B e. [ B ] .~ )
85		simpr 458	. . . . . . . . 9 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> z = ( A D B ) )
86		oveq1 6097	. . . . . . . . . . 11 |- ( a = A -> ( a D b ) = ( A D b ) )
87	86	eqeq2d 2452	. . . . . . . . . 10 |- ( a = A -> ( z = ( a D b ) <-> z = ( A D b ) ) )
88		oveq2 6098	. . . . . . . . . . 11 |- ( b = B -> ( A D b ) = ( A D B ) )
89	88	eqeq2d 2452	. . . . . . . . . 10 |- ( b = B -> ( z = ( A D b ) <-> z = ( A D B ) ) )
90	87, 89	rspc2ev 3078	. . . . . . . . 9 |- ( ( A e. [ A ] .~ /\ B e. [ B ] .~ /\ z = ( A D B ) ) -> E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) )
91	74, 84, 85, 90	syl3anc 1213	. . . . . . . 8 |- ( ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) /\ z = ( A D B ) ) -> E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) )
92	91	ex 434	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( z = ( A D B ) -> E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) ) )
93	62, 92	impbid 191	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) <-> z = ( A D B ) ) )
94	93	abbidv 2555	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> { z | E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) } = { z | z = ( A D B ) } )
95		df-sn 3875	. . . . 5 |- { ( A D B ) } = { z | z = ( A D B ) }
96	94, 95	syl6eqr 2491	. . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> { z | E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) } = { ( A D B ) } )
97	96	unieqd 4098	. . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> U. { z | E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) } = U. { ( A D B ) } )
98		ovex 6115	. . . 4 |- ( A D B ) e. _V
99	98	unisn 4103	. . 3 |- U. { ( A D B ) } = ( A D B )
100	97, 99	syl6eq 2489	. 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> U. { z | E. a e. [ A ] .~ E. b e. [ B ] .~ z = ( a D b ) } = ( A D B ) )
101	4, 26, 100	3eqtrd 2477	1 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( [ A ] .~ (pstoMet ` D ) [ B ] .~ ) = ( A D B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 960   = wceq 1364   e. wcel 1761   {cab 2427   E.wrex 2714   _Vcvv 2970   C_ wss 3325   {csn 3874   U.cuni 4088   class class class wbr 4289   X. cxp 4834   Rel wrel 4841   ` cfv 5415  (class class class)co 6090   e. cmpt2 6092   [cec 7095   /.cqs 7096   0cc0 9278  PsMetcpsmet 17759  ~Metcmetid 26249  pstoMetcpstm 26250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-xadd 11086  df-psmet 17768  df-metid 26251  df-pstm 26252

This theorem is referenced by:  pstmxmet  26260