txbas - Metamath Proof Explorer

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Theorem txbas 21370

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 5128	. . . . . . . . . 10
3		xpeq2 5129	. . . . . . . . . 10
4	2, 3	cbvmpt2v 6735	. . . . . . . . 9
5	4	rnmpt2 6770	. . . . . . . 8
6	1, 5	eqtri 2644	. . . . . . 7
7	6	abeq2i 2735	. . . . . 6
8		xpeq1 5128	. . . . . . . . . 10
9		xpeq2 5129	. . . . . . . . . 10
10	8, 9	cbvmpt2v 6735	. . . . . . . . 9
11	10	rnmpt2 6770	. . . . . . . 8
12	1, 11	eqtri 2644	. . . . . . 7
13	12	abeq2i 2735	. . . . . 6
14	7, 13	anbi12i 733	. . . . 5 $/\$ $/\$
15		reeanv 3107	. . . . 5 $/\$ $/\$
16	14, 15	bitr4i 267	. . . 4 $/\$ $/\$
17		reeanv 3107	. . . . . 6 $/\$ $/\$
18		basis2 20755	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
19	18	exp43 640	. . . . . . . . . . . . . . 15 $/\$
20	19	imp42 620	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
21		basis2 20755	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
22	21	exp43 640	. . . . . . . . . . . . . . 15 $/\$
23	22	imp42 620	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
24		reeanv 3107	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25		opelxpi 5148	. . . . . . . . . . . . . . . . . . 19 $/\$
26		xpss12 5225	. . . . . . . . . . . . . . . . . . 19 $/\$
27	25, 26	anim12i 590	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$
28	27	an4s 869	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
29	28	reximi 3011	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
30	29	reximi 3011	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
31	24, 30	sylbir 225	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
32	20, 23, 31	syl2an 494	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	an4s 869	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	33	ralrimivva 2971	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		eleq1 2689	. . . . . . . . . . . . . 14
36	35	anbi1d 741	. . . . . . . . . . . . 13 $/\$ $/\$
37	36	2rexbidv 3057	. . . . . . . . . . . 12 $/\$ $/\$
38	37	ralxp 5263	. . . . . . . . . . 11 $/\$ $/\$
39	34, 38	sylibr 224	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	39	an4s 869	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	40	anassrs 680	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		ineq12 3809	. . . . . . . . . 10 $/\$
43		inxp 5254	. . . . . . . . . 10
44	42, 43	syl6eq 2672	. . . . . . . . 9 $/\$
45	44	sseq2d 3633	. . . . . . . . . . . 12 $/\$
46	45	anbi2d 740	. . . . . . . . . . 11 $/\$ $/\$ $/\$
47	46	rexbidv 3052	. . . . . . . . . 10 $/\$ $/\$ $/\$
48	1	rexeqi 3143	. . . . . . . . . . 11 $/\$ $/\$
49		fvex 6201	. . . . . . . . . . . . . 14
50		fvex 6201	. . . . . . . . . . . . . 14
51	49, 50	xpex 6962	. . . . . . . . . . . . 13
52	51	rgenw 2924	. . . . . . . . . . . 12
53		vex 3203	. . . . . . . . . . . . . . . . 17
54		vex 3203	. . . . . . . . . . . . . . . . 17
55	53, 54	op1std 7178	. . . . . . . . . . . . . . . 16
56	53, 54	op2ndd 7179	. . . . . . . . . . . . . . . 16
57	55, 56	xpeq12d 5140	. . . . . . . . . . . . . . 15
58	57	mpt2mpt 6752	. . . . . . . . . . . . . 14
59	58	eqcomi 2631	. . . . . . . . . . . . 13
60		eleq2 2690	. . . . . . . . . . . . . 14
61		sseq1 3626	. . . . . . . . . . . . . 14
62	60, 61	anbi12d 747	. . . . . . . . . . . . 13 $/\$ $/\$
63	59, 62	rexrnmpt 6369	. . . . . . . . . . . 12 $/\$ $/\$
64	52, 63	ax-mp 5	. . . . . . . . . . 11 $/\$ $/\$
65	57	eleq2d 2687	. . . . . . . . . . . . 13
66	57	sseq1d 3632	. . . . . . . . . . . . 13
67	65, 66	anbi12d 747	. . . . . . . . . . . 12 $/\$ $/\$
68	67	rexxp 5264	. . . . . . . . . . 11 $/\$ $/\$
69	48, 64, 68	3bitri 286	. . . . . . . . . 10 $/\$ $/\$
70	47, 69	syl6bb 276	. . . . . . . . 9 $/\$ $/\$ $/\$
71	44, 70	raleqbidv 3152	. . . . . . . 8 $/\$ $/\$ $/\$
72	41, 71	syl5ibrcom 237	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73	72	rexlimdvva 3038	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
74	17, 73	syl5bir 233	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
75	74	rexlimdvva 3038	. . . 4 $/\$ $/\$ $/\$
76	16, 75	syl5bi 232	. . 3 $/\$ $/\$ $/\$
77	76	ralrimivv 2970	. 2 $/\$ $/\$
78	1	txbasex 21369	. . 3 $/\$
79		isbasis2g 20752	. . 3 $/\$
80	78, 79	syl 17	. 2 $/\$ $/\$
81	77, 80	mpbird 247	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

cab 2608

wral 2912

wrex 2913

cvv 3200

cin 3573

wss 3574

cop 4183

cmpt 4729

cxp 5112

crn 5115

cfv 5888

cmpt2 6652

c1st 7166

c2nd 7167

ctb 20749

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-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-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-fv 5896 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-bases 20750

This theorem is referenced by: txtop 21372 tx2ndc 21454 mbfimaopnlem 23422

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