Theorem ntrclsiso 38365

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that either is isotonic hold equally. (Contributed by RP, 3-Jun-2021.)

Hypotheses

Ref	Expression
ntrcls.o	|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
ntrcls.d	|- D = ( O ` B )
ntrcls.r	|- ( ph -> I D K )

Assertion

Ref	Expression
ntrclsiso	|- ( ph -> ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> A. s e. ~P B A. t e. ~P B ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) ) )

Distinct variable groups:   B, i, j, k, s, t   j, I, k, s, t   ph, i, j, k, s, t

Allowed substitution hints:   D( t, i, j, k, s)   I( i)   K( t, i, j, k, s)   O( t, i, j, k, s)

Proof of Theorem ntrclsiso

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3626	. . . . 5 |- ( s = b -> ( s C_ t <-> b C_ t ) )
2		fveq2 6191	. . . . . 6 |- ( s = b -> ( I ` s ) = ( I ` b ) )
3	2	sseq1d 3632	. . . . 5 |- ( s = b -> ( ( I ` s ) C_ ( I ` t ) <-> ( I ` b ) C_ ( I ` t ) ) )
4	1, 3	imbi12d 334	. . . 4 |- ( s = b -> ( ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> ( b C_ t -> ( I ` b ) C_ ( I ` t ) ) ) )
5		sseq2 3627	. . . . 5 |- ( t = a -> ( b C_ t <-> b C_ a ) )
6		fveq2 6191	. . . . . 6 |- ( t = a -> ( I ` t ) = ( I ` a ) )
7	6	sseq2d 3633	. . . . 5 |- ( t = a -> ( ( I ` b ) C_ ( I ` t ) <-> ( I ` b ) C_ ( I ` a ) ) )
8	5, 7	imbi12d 334	. . . 4 |- ( t = a -> ( ( b C_ t -> ( I ` b ) C_ ( I ` t ) ) <-> ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) ) )
9	4, 8	cbvral2v 3179	. . 3 |- ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> A. b e. ~P B A. a e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) )
10		ralcom 3098	. . 3 |- ( A. b e. ~P B A. a e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) <-> A. a e. ~P B A. b e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) )
11	9, 10	bitri 264	. 2 |- ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> A. a e. ~P B A. b e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) )
12		simpl 473	. . . . 5 |- ( ( ph /\ s e. ~P B ) -> ph )
13		ntrcls.d	. . . . . 6 |- D = ( O ` B )
14		ntrcls.r	. . . . . 6 |- ( ph -> I D K )
15	13, 14	ntrclsbex 38332	. . . . 5 |- ( ph -> B e. _V )
16	12, 15	syl 17	. . . 4 |- ( ( ph /\ s e. ~P B ) -> B e. _V )
17		difssd 3738	. . . 4 |- ( ( ph /\ s e. ~P B ) -> ( B \ s ) C_ B )
18	16, 17	sselpwd 4807	. . 3 |- ( ( ph /\ s e. ~P B ) -> ( B \ s ) e. ~P B )
19		elpwi 4168	. . . 4 |- ( a e. ~P B -> a C_ B )
20		simpl 473	. . . . . 6 |- ( ( B e. _V /\ a C_ B ) -> B e. _V )
21		difssd 3738	. . . . . 6 |- ( ( B e. _V /\ a C_ B ) -> ( B \ a ) C_ B )
22	20, 21	sselpwd 4807	. . . . 5 |- ( ( B e. _V /\ a C_ B ) -> ( B \ a ) e. ~P B )
23		simpr 477	. . . . . . . 8 |- ( ( ( B e. _V /\ a C_ B ) /\ s = ( B \ a ) ) -> s = ( B \ a ) )
24	23	difeq2d 3728	. . . . . . 7 |- ( ( ( B e. _V /\ a C_ B ) /\ s = ( B \ a ) ) -> ( B \ s ) = ( B \ ( B \ a ) ) )
25	24	eqeq2d 2632	. . . . . 6 |- ( ( ( B e. _V /\ a C_ B ) /\ s = ( B \ a ) ) -> ( a = ( B \ s ) <-> a = ( B \ ( B \ a ) ) ) )
26		eqcom 2629	. . . . . 6 |- ( a = ( B \ ( B \ a ) ) <-> ( B \ ( B \ a ) ) = a )
27	25, 26	syl6bb 276	. . . . 5 |- ( ( ( B e. _V /\ a C_ B ) /\ s = ( B \ a ) ) -> ( a = ( B \ s ) <-> ( B \ ( B \ a ) ) = a ) )
28		dfss4 3858	. . . . . . 7 |- ( a C_ B <-> ( B \ ( B \ a ) ) = a )
29	28	biimpi 206	. . . . . 6 |- ( a C_ B -> ( B \ ( B \ a ) ) = a )
30	29	adantl 482	. . . . 5 |- ( ( B e. _V /\ a C_ B ) -> ( B \ ( B \ a ) ) = a )
31	22, 27, 30	rspcedvd 3317	. . . 4 |- ( ( B e. _V /\ a C_ B ) -> E. s e. ~P B a = ( B \ s ) )
32	15, 19, 31	syl2an 494	. . 3 |- ( ( ph /\ a e. ~P B ) -> E. s e. ~P B a = ( B \ s ) )
33		simpl1 1064	. . . . . 6 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B ) -> ph )
34	33, 15	syl 17	. . . . 5 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B ) -> B e. _V )
35		difssd 3738	. . . . 5 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B ) -> ( B \ t ) C_ B )
36	34, 35	sselpwd 4807	. . . 4 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B ) -> ( B \ t ) e. ~P B )
37		elpwi 4168	. . . . . 6 |- ( b e. ~P B -> b C_ B )
38		simpl 473	. . . . . . . 8 |- ( ( B e. _V /\ b C_ B ) -> B e. _V )
39		difssd 3738	. . . . . . . 8 |- ( ( B e. _V /\ b C_ B ) -> ( B \ b ) C_ B )
40	38, 39	sselpwd 4807	. . . . . . 7 |- ( ( B e. _V /\ b C_ B ) -> ( B \ b ) e. ~P B )
41		simpr 477	. . . . . . . . . 10 |- ( ( ( B e. _V /\ b C_ B ) /\ t = ( B \ b ) ) -> t = ( B \ b ) )
42	41	difeq2d 3728	. . . . . . . . 9 |- ( ( ( B e. _V /\ b C_ B ) /\ t = ( B \ b ) ) -> ( B \ t ) = ( B \ ( B \ b ) ) )
43	42	eqeq2d 2632	. . . . . . . 8 |- ( ( ( B e. _V /\ b C_ B ) /\ t = ( B \ b ) ) -> ( b = ( B \ t ) <-> b = ( B \ ( B \ b ) ) ) )
44		eqcom 2629	. . . . . . . 8 |- ( b = ( B \ ( B \ b ) ) <-> ( B \ ( B \ b ) ) = b )
45	43, 44	syl6bb 276	. . . . . . 7 |- ( ( ( B e. _V /\ b C_ B ) /\ t = ( B \ b ) ) -> ( b = ( B \ t ) <-> ( B \ ( B \ b ) ) = b ) )
46		dfss4 3858	. . . . . . . . 9 |- ( b C_ B <-> ( B \ ( B \ b ) ) = b )
47	46	biimpi 206	. . . . . . . 8 |- ( b C_ B -> ( B \ ( B \ b ) ) = b )
48	47	adantl 482	. . . . . . 7 |- ( ( B e. _V /\ b C_ B ) -> ( B \ ( B \ b ) ) = b )
49	40, 45, 48	rspcedvd 3317	. . . . . 6 |- ( ( B e. _V /\ b C_ B ) -> E. t e. ~P B b = ( B \ t ) )
50	15, 37, 49	syl2an 494	. . . . 5 |- ( ( ph /\ b e. ~P B ) -> E. t e. ~P B b = ( B \ t ) )
51	50	3ad2antl1 1223	. . . 4 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ b e. ~P B ) -> E. t e. ~P B b = ( B \ t ) )
52		simp12 1092	. . . . . . . . 9 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> s e. ~P B )
53	52	elpwid 4170	. . . . . . . 8 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> s C_ B )
54		simp2 1062	. . . . . . . . 9 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> t e. ~P B )
55	54	elpwid 4170	. . . . . . . 8 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> t C_ B )
56		sscon34b 38317	. . . . . . . 8 |- ( ( s C_ B /\ t C_ B ) -> ( s C_ t <-> ( B \ t ) C_ ( B \ s ) ) )
57	53, 55, 56	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( s C_ t <-> ( B \ t ) C_ ( B \ s ) ) )
58	57	bicomd 213	. . . . . 6 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( B \ t ) C_ ( B \ s ) <-> s C_ t ) )
59		simp11 1091	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ph )
60		ntrcls.o	. . . . . . . . . . . 12 |- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
61	60, 13, 14	ntrclsiex 38351	. . . . . . . . . . 11 |- ( ph -> I e. ( ~P B ^m ~P B ) )
62	59, 61	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> I e. ( ~P B ^m ~P B ) )
63		elmapi 7879	. . . . . . . . . 10 |- ( I e. ( ~P B ^m ~P B ) -> I : ~P B --> ~P B )
64	62, 63	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> I : ~P B --> ~P B )
65	59, 15	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> B e. _V )
66		difssd 3738	. . . . . . . . . 10 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( B \ t ) C_ B )
67	65, 66	sselpwd 4807	. . . . . . . . 9 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( B \ t ) e. ~P B )
68	64, 67	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` ( B \ t ) ) e. ~P B )
69	68	elpwid 4170	. . . . . . 7 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` ( B \ t ) ) C_ B )
70		difssd 3738	. . . . . . . . . 10 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( B \ s ) C_ B )
71	65, 70	sselpwd 4807	. . . . . . . . 9 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( B \ s ) e. ~P B )
72	64, 71	ffvelrnd 6360	. . . . . . . 8 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` ( B \ s ) ) e. ~P B )
73	72	elpwid 4170	. . . . . . 7 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` ( B \ s ) ) C_ B )
74		sscon34b 38317	. . . . . . 7 |- ( ( ( I ` ( B \ t ) ) C_ B /\ ( I ` ( B \ s ) ) C_ B ) -> ( ( I ` ( B \ t ) ) C_ ( I ` ( B \ s ) ) <-> ( B \ ( I ` ( B \ s ) ) ) C_ ( B \ ( I ` ( B \ t ) ) ) ) )
75	69, 73, 74	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( I ` ( B \ t ) ) C_ ( I ` ( B \ s ) ) <-> ( B \ ( I ` ( B \ s ) ) ) C_ ( B \ ( I ` ( B \ t ) ) ) ) )
76	58, 75	imbi12d 334	. . . . 5 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( ( B \ t ) C_ ( B \ s ) -> ( I ` ( B \ t ) ) C_ ( I ` ( B \ s ) ) ) <-> ( s C_ t -> ( B \ ( I ` ( B \ s ) ) ) C_ ( B \ ( I ` ( B \ t ) ) ) ) ) )
77		simp3 1063	. . . . . . 7 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> b = ( B \ t ) )
78		simp13 1093	. . . . . . 7 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> a = ( B \ s ) )
79	77, 78	sseq12d 3634	. . . . . 6 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( b C_ a <-> ( B \ t ) C_ ( B \ s ) ) )
80	77	fveq2d 6195	. . . . . . 7 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` b ) = ( I ` ( B \ t ) ) )
81	78	fveq2d 6195	. . . . . . 7 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( I ` a ) = ( I ` ( B \ s ) ) )
82	80, 81	sseq12d 3634	. . . . . 6 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( I ` b ) C_ ( I ` a ) <-> ( I ` ( B \ t ) ) C_ ( I ` ( B \ s ) ) ) )
83	79, 82	imbi12d 334	. . . . 5 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) <-> ( ( B \ t ) C_ ( B \ s ) -> ( I ` ( B \ t ) ) C_ ( I ` ( B \ s ) ) ) ) )
84	60, 13, 14	ntrclsfv1 38353	. . . . . . . . . 10 |- ( ph -> ( D ` I ) = K )
85	59, 84	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( D ` I ) = K )
86	85	fveq1d 6193	. . . . . . . 8 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( D ` I ) ` s ) = ( K ` s ) )
87		eqid 2622	. . . . . . . . 9 |- ( D ` I ) = ( D ` I )
88		eqid 2622	. . . . . . . . 9 |- ( ( D ` I ) ` s ) = ( ( D ` I ) ` s )
89	60, 13, 65, 62, 87, 52, 88	dssmapfv3d 38313	. . . . . . . 8 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( D ` I ) ` s ) = ( B \ ( I ` ( B \ s ) ) ) )
90	86, 89	eqtr3d 2658	. . . . . . 7 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( K ` s ) = ( B \ ( I ` ( B \ s ) ) ) )
91	59, 14	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> I D K )
92	60, 13, 91	ntrclsfv1 38353	. . . . . . . . 9 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( D ` I ) = K )
93	92	fveq1d 6193	. . . . . . . 8 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( D ` I ) ` t ) = ( K ` t ) )
94		eqid 2622	. . . . . . . . 9 |- ( ( D ` I ) ` t ) = ( ( D ` I ) ` t )
95	60, 13, 65, 62, 87, 54, 94	dssmapfv3d 38313	. . . . . . . 8 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( D ` I ) ` t ) = ( B \ ( I ` ( B \ t ) ) ) )
96	93, 95	eqtr3d 2658	. . . . . . 7 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( K ` t ) = ( B \ ( I ` ( B \ t ) ) ) )
97	90, 96	sseq12d 3634	. . . . . 6 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( K ` s ) C_ ( K ` t ) <-> ( B \ ( I ` ( B \ s ) ) ) C_ ( B \ ( I ` ( B \ t ) ) ) ) )
98	97	imbi2d 330	. . . . 5 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) <-> ( s C_ t -> ( B \ ( I ` ( B \ s ) ) ) C_ ( B \ ( I ` ( B \ t ) ) ) ) ) )
99	76, 83, 98	3bitr4d 300	. . . 4 |- ( ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) /\ t e. ~P B /\ b = ( B \ t ) ) -> ( ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) <-> ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) ) )
100	36, 51, 99	ralxfrd2 4884	. . 3 |- ( ( ph /\ s e. ~P B /\ a = ( B \ s ) ) -> ( A. b e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) <-> A. t e. ~P B ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) ) )
101	18, 32, 100	ralxfrd2 4884	. 2 |- ( ph -> ( A. a e. ~P B A. b e. ~P B ( b C_ a -> ( I ` b ) C_ ( I ` a ) ) <-> A. s e. ~P B A. t e. ~P B ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) ) )
102	11, 101	syl5bb 272	1 |- ( ph -> ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> A. s e. ~P B A. t e. ~P B ( s C_ t -> ( K ` s ) C_ ( K ` t ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by: (None)