retopbas - Metamath Proof Explorer

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Theorem retopbas 24781

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 13487	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2	1	fdmi 6747	. . . 4 ⊢ dom (,) = (ℝ^* × ℝ^*)
3	2	imaeq2i 6076	. . 3 ⊢ ((,) “ dom (,)) = ((,) “ (ℝ^* × ℝ^*))
4		imadmrn 6088	. . 3 ⊢ ((,) “ dom (,)) = ran (,)
5	3, 4	eqtr3i 2767	. 2 ⊢ ((,) “ (ℝ^* × ℝ^*)) = ran (,)
6		ssid 4006	. . 3 ⊢ ℝ^* ⊆ ℝ^*
7	6	qtopbaslem 24779	. 2 ⊢ ((,) “ (ℝ^* × ℝ^*)) ∈ TopBases
8	5, 7	eqeltrri 2838	1 ⊢ ran (,) ∈ TopBases

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2108 𝒫 cpw 4600 × cxp 5683 dom cdm 5685 ran crn 5686 “ cima 5688 ℝcr 11154 ℝ^*cxr 11294 (,)cioo 13387 TopBasesctb 22952

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-bases 22953

This theorem is referenced by: retop 24782 uniretop 24783 iooretop 24786 qdensere 24790 tgioo 24817 xrtgioo 24828 bndth 24990 ovolicc2 25557 cncombf 25693 cnmbf 25694 elmbfmvol2 34269 iccllysconn 35255 rellysconn 35256 mblfinlem3 37666 mblfinlem4 37667 ismblfin 37668 cnambfre 37675