tpspropd - Metamath Proof Explorer

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Theorem tpspropd 22854

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 6832	. . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿)))
4	1, 3	eleq12d 2827	. 2 ⊢ (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))))
5		eqid 2733	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2733	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
7	5, 6	istps 22850	. 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
8		eqid 2733	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
9		eqid 2733	. . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿)
10	8, 9	istps 22850	. 2 ⊢ (𝐿 ∈ TopSp ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿)))
11	4, 7, 10	3bitr4g 314	1 ⊢ (𝜑 → (𝐾 ∈ TopSp ↔ 𝐿 ∈ TopSp))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 = wceq 1541 ∈ wcel 2113 ‘cfv 6486 Basecbs 17122 TopOpenctopn 17327 TopOnctopon 22826 TopSpctps 22848

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-iota 6442 df-fun 6488 df-fv 6494 df-top 22810 df-topon 22827 df-topsp 22849

This theorem is referenced by: tpsprop2d 22855 xmspropd 24389

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