tpspropd - Metamath Proof Explorer

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Theorem tpspropd 21543

Description: A topological space depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 13-Aug-2015.)

**Hypotheses**
Ref	Expression
tpspropd.1	⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
tpspropd.2	⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

**Assertion**
Ref	Expression
tpspropd	⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

**Proof of Theorem tpspropd**
Step	Hyp	Ref	Expression
1		tpspropd.2	. . 3 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))
2		tpspropd.1	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
3	2	fveq2d 6649	. . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿)))
4	1, 3	eleq12d 2884	. 2 ⊢ (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))))
5		eqid 2798	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2798	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
7	5, 6	istps 21539	. 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
8		eqid 2798	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
9		eqid 2798	. . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿)
10	8, 9	istps 21539	. 2 ⊢ (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))
11	4, 7, 10	3bitr4g 317	1 ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 Basecbs 16475 TopOpenctopn 16687 TopOnctopon 21515 TopSpctps 21537

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-top 21499 df-topon 21516 df-topsp 21538

This theorem is referenced by: tpsprop2d 21544 xmspropd 23080

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