retopbas - Metamath Proof Explorer

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Theorem retopbas 24796

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 13483	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2	1	fdmi 6747	. . . 4 ⊢ dom (,) = (ℝ^* × ℝ^*)
3	2	imaeq2i 6077	. . 3 ⊢ ((,) “ dom (,)) = ((,) “ (ℝ^* × ℝ^*))
4		imadmrn 6089	. . 3 ⊢ ((,) “ dom (,)) = ran (,)
5	3, 4	eqtr3i 2764	. 2 ⊢ ((,) “ (ℝ^* × ℝ^*)) = ran (,)
6		ssid 4017	. . 3 ⊢ ℝ^* ⊆ ℝ^*
7	6	qtopbaslem 24794	. 2 ⊢ ((,) “ (ℝ^* × ℝ^*)) ∈ TopBases
8	5, 7	eqeltrri 2835	1 ⊢ ran (,) ∈ TopBases

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2105 𝒫 cpw 4604 × cxp 5686 dom cdm 5688 ran crn 5689 “ cima 5691 ℝcr 11151 ℝ^*cxr 11291 (,)cioo 13383 TopBasesctb 22967

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-pre-lttri 11226 ax-pre-lttrn 11227

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-1st 8012 df-2nd 8013 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-ioo 13387 df-bases 22968

This theorem is referenced by: retop 24797 uniretop 24798 iooretop 24801 qdensere 24805 tgioo 24831 xrtgioo 24841 bndth 25003 ovolicc2 25570 cncombf 25706 cnmbf 25707 elmbfmvol2 34248 iccllysconn 35234 rellysconn 35235 mblfinlem3 37645 mblfinlem4 37646 ismblfin 37647 cnambfre 37654