retopbas - Metamath Proof Explorer

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Theorem retopbas 23371

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 12838	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
2	1	fdmi 6526	. . . 4 ⊢ dom (,) = (ℝ^* × ℝ^*)
3	2	imaeq2i 5929	. . 3 ⊢ ((,) “ dom (,)) = ((,) “ (ℝ^* × ℝ^*))
4		imadmrn 5941	. . 3 ⊢ ((,) “ dom (,)) = ran (,)
5	3, 4	eqtr3i 2848	. 2 ⊢ ((,) “ (ℝ^* × ℝ^*)) = ran (,)
6		ssid 3991	. . 3 ⊢ ℝ^* ⊆ ℝ^*
7	6	qtopbaslem 23369	. 2 ⊢ ((,) “ (ℝ^* × ℝ^*)) ∈ TopBases
8	5, 7	eqeltrri 2912	1 ⊢ ran (,) ∈ TopBases

Colors of variables: wff setvar class

Syntax hints: ∈ wcel 2114 𝒫 cpw 4541 × cxp 5555 dom cdm 5557 ran crn 5558 “ cima 5560 ℝcr 10538 ℝ^*cxr 10676 (,)cioo 12741 TopBasesctb 21555

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ioo 12745 df-bases 21556

This theorem is referenced by: retop 23372 uniretop 23373 iooretop 23376 qdensere 23380 tgioo 23406 xrtgioo 23416 bndth 23564 ovolicc2 24125 cncombf 24261 cnmbf 24262 elmbfmvol2 31527 iccllysconn 32499 rellysconn 32500 mblfinlem3 34933 mblfinlem4 34934 ismblfin 34935 cnambfre 34942