txbas - Intuitionistic Logic Explorer

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Theorem txbas 13728

Description: The set of Cartesian products of elements from two topological bases is a basis. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 31-Aug-2015.)

**Hypothesis**
Ref	Expression
txval.1

**Assertion**
Ref	Expression
txbas	$/\$

Distinct variable groups:

Allowed substitution hints:

(

)

Proof of Theorem txbas Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		txval.1	. . . . . . . 8
2		xpeq1 4640	. . . . . . . . . 10
3		xpeq2 4641	. . . . . . . . . 10
4	2, 3	cbvmpov 5954	. . . . . . . . 9
5	4	rnmpo 5984	. . . . . . . 8
6	1, 5	eqtri 2198	. . . . . . 7
7	6	abeq2i 2288	. . . . . 6
8		xpeq1 4640	. . . . . . . . . 10
9		xpeq2 4641	. . . . . . . . . 10
10	8, 9	cbvmpov 5954	. . . . . . . . 9
11	10	rnmpo 5984	. . . . . . . 8
12	1, 11	eqtri 2198	. . . . . . 7
13	12	abeq2i 2288	. . . . . 6
14	7, 13	anbi12i 460	. . . . 5 $/\$ $/\$
15		reeanv 2646	. . . . 5 $/\$ $/\$
16	14, 15	bitr4i 187	. . . 4 $/\$ $/\$
17		reeanv 2646	. . . . . 6 $/\$ $/\$
18		basis2 13518	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
19	18	exp43 372	. . . . . . . . . . . . . . 15 $/\$
20	19	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
21		basis2 13518	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
22	21	exp43 372	. . . . . . . . . . . . . . 15 $/\$
23	22	imp42 354	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
24		reeanv 2646	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		opelxpi 4658	. . . . . . . . . . . . . . . . . . 19 $/\$
26		xpss12 4733	. . . . . . . . . . . . . . . . . . 19 $/\$
27	25, 26	anim12i 338	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
28	27	an4s 588	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
29	28	reximi 2574	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
30	29	reximi 2574	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
31	24, 30	sylbir 135	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
32	20, 23, 31	syl2an 289	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	an4s 588	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	ralrimivva 2559	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		eleq1 2240	. . . . . . . . . . . . . 14
36	35	anbi1d 465	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	2rexbidv 2502	. . . . . . . . . . . 12 $/\$ $/\$
38	37	ralxp 4770	. . . . . . . . . . 11 $/\$ $/\$
39	34, 38	sylibr 134	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an4s 588	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	anassrs 400	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		ineq12 3331	. . . . . . . . . 10 $/\$
43		inxp 4761	. . . . . . . . . 10
44	42, 43	eqtrdi 2226	. . . . . . . . 9 $/\$
45	44	sseq2d 3185	. . . . . . . . . . . 12 $/\$
46	45	anbi2d 464	. . . . . . . . . . 11 $/\$ $/\$ $/\$
47	46	rexbidv 2478	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	1	rexeqi 2677	. . . . . . . . . . 11 $/\$ $/\$
49		1stexg 6167	. . . . . . . . . . . . . . 15
50	49	elv 2741	. . . . . . . . . . . . . 14
51		2ndexg 6168	. . . . . . . . . . . . . . 15
52	51	elv 2741	. . . . . . . . . . . . . 14
53	50, 52	xpex 4741	. . . . . . . . . . . . 13
54	53	rgenw 2532	. . . . . . . . . . . 12
55		vex 2740	. . . . . . . . . . . . . . . . 17
56		vex 2740	. . . . . . . . . . . . . . . . 17
57	55, 56	op1std 6148	. . . . . . . . . . . . . . . 16
58	55, 56	op2ndd 6149	. . . . . . . . . . . . . . . 16
59	57, 58	xpeq12d 4651	. . . . . . . . . . . . . . 15
60	59	mpompt 5966	. . . . . . . . . . . . . 14
61	60	eqcomi 2181	. . . . . . . . . . . . 13
62		eleq2 2241	. . . . . . . . . . . . . 14
63		sseq1 3178	. . . . . . . . . . . . . 14
64	62, 63	anbi12d 473	. . . . . . . . . . . . 13 $/\$ $/\$
65	61, 64	rexrnmpt 5659	. . . . . . . . . . . 12 $/\$ $/\$
66	54, 65	ax-mp 5	. . . . . . . . . . 11 $/\$ $/\$
67	59	eleq2d 2247	. . . . . . . . . . . . 13
68	59	sseq1d 3184	. . . . . . . . . . . . 13
69	67, 68	anbi12d 473	. . . . . . . . . . . 12 $/\$ $/\$
70	69	rexxp 4771	. . . . . . . . . . 11 $/\$ $/\$
71	48, 66, 70	3bitri 206	. . . . . . . . . 10 $/\$ $/\$
72	47, 71	bitrdi 196	. . . . . . . . 9 $/\$ $/\$ $/\$
73	44, 72	raleqbidv 2684	. . . . . . . 8 $/\$ $/\$ $/\$
74	41, 73	syl5ibrcom 157	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	74	rexlimdvva 2602	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
76	17, 75	biimtrrid 153	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77	76	rexlimdvva 2602	. . . 4 $/\$ $/\$ $/\$
78	16, 77	biimtrid 152	. . 3 $/\$ $/\$ $/\$
79	78	ralrimivv 2558	. 2 $/\$ $/\$
80	1	txbasex 13727	. . 3 $/\$
81		isbasis2g 13515	. . 3 $/\$
82	80, 81	syl 14	. 2 $/\$ $/\$
83	79, 82	mpbird 167	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1353

wcel 2148

cab 2163

wral 2455

wrex 2456

cvv 2737

cin 3128

wss 3129

cop 3595

cmpt 4064

cxp 4624

crn 4627

cfv 5216

cmpo 5876

c1st 6138

c2nd 6139

ctb 13512

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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-fo 5222 df-fv 5224 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-bases 13513

This theorem is referenced by: txtop 13730

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