retopbas - Metamath Proof Explorer

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Theorem retopbas 24655

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 13415	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2	1	fdmi 6702	. . . 4 ⊢ dom (,) = (ℝ^* × ℝ^*)
3	2	imaeq2i 6032	. . 3 ⊢ ((,) “ dom (,)) = ((,) “ (ℝ^* × ℝ^*))
4		imadmrn 6044	. . 3 ⊢ ((,) “ dom (,)) = ran (,)
5	3, 4	eqtr3i 2755	. 2 ⊢ ((,) “ (ℝ^* × ℝ^*)) = ran (,)
6		ssid 3972	. . 3 ⊢ ℝ^* ⊆ ℝ^*
7	6	qtopbaslem 24653	. 2 ⊢ ((,) “ (ℝ^* × ℝ^*)) ∈ TopBases
8	5, 7	eqeltrri 2826	1 ⊢ ran (,) ∈ TopBases

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2109 𝒫 cpw 4566 × cxp 5639 dom cdm 5641 ran crn 5642 “ cima 5644 ℝcr 11074 ℝ^*cxr 11214 (,)cioo 13313 TopBasesctb 22839

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-pre-lttri 11149 ax-pre-lttrn 11150

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-ioo 13317 df-bases 22840

This theorem is referenced by: retop 24656 uniretop 24657 iooretop 24660 qdensere 24664 tgioo 24691 xrtgioo 24702 bndth 24864 ovolicc2 25430 cncombf 25566 cnmbf 25567 elmbfmvol2 34265 iccllysconn 35244 rellysconn 35245 mblfinlem3 37660 mblfinlem4 37661 ismblfin 37662 cnambfre 37669