tpspropd - Metamath Proof Explorer

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Theorem tpspropd 21995

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 6760	. . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿)))
4	1, 3	eleq12d 2833	. 2 ⊢ (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))))
5		eqid 2738	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2738	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
7	5, 6	istps 21991	. 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
8		eqid 2738	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
9		eqid 2738	. . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿)
10	8, 9	istps 21991	. 2 ⊢ (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))
11	4, 7, 10	3bitr4g 313	1 ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 Basecbs 16840 TopOpenctopn 17049 TopOnctopon 21967 TopSpctps 21989

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-iota 6376 df-fun 6420 df-fv 6426 df-top 21951 df-topon 21968 df-topsp 21990

This theorem is referenced by: tpsprop2d 21996 xmspropd 23534

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