retopbas - Metamath Proof Explorer

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Theorem retopbas 24735

Description: A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)

**Assertion**
Ref	Expression
retopbas	⊢ ran (,) ∈ TopBases

**Proof of Theorem retopbas**
Step	Hyp	Ref	Expression
1		ioof 13391	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2	1	fdmi 6673	. . . 4 ⊢ dom (,) = (ℝ^* × ℝ^*)
3	2	imaeq2i 6017	. . 3 ⊢ ((,) “ dom (,)) = ((,) “ (ℝ^* × ℝ^*))
4		imadmrn 6029	. . 3 ⊢ ((,) “ dom (,)) = ran (,)
5	3, 4	eqtr3i 2762	. 2 ⊢ ((,) “ (ℝ^* × ℝ^*)) = ran (,)
6		ssid 3945	. . 3 ⊢ ℝ^* ⊆ ℝ^*
7	6	qtopbaslem 24733	. 2 ⊢ ((,) “ (ℝ^* × ℝ^*)) ∈ TopBases
8	5, 7	eqeltrri 2834	1 ⊢ ran (,) ∈ TopBases

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2114 𝒫 cpw 4542 × cxp 5622 dom cdm 5624 ran crn 5625 “ cima 5627 ℝcr 11028 ℝ^*cxr 11169 (,)cioo 13289 TopBasesctb 22920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-pre-lttri 11103 ax-pre-lttrn 11104

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-ioo 13293 df-bases 22921

This theorem is referenced by: retop 24736 uniretop 24737 iooretop 24740 qdensere 24744 tgioo 24771 xrtgioo 24782 bndth 24935 ovolicc2 25499 cncombf 25635 cnmbf 25636 elmbfmvol2 34427 iccllysconn 35448 rellysconn 35449 mblfinlem3 37994 mblfinlem4 37995 ismblfin 37996 cnambfre 38003