tpspropd - Metamath Proof Explorer

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Theorem tpspropd 22965

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 6924	. . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿)))
4	1, 3	eleq12d 2838	. 2 ⊢ (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))))
5		eqid 2740	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2740	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
7	5, 6	istps 22961	. 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
8		eqid 2740	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
9		eqid 2740	. . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿)
10	8, 9	istps 22961	. 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 1537 ∈ wcel 2108 ‘cfv 6573 Basecbs 17258 TopOpenctopn 17481 TopOnctopon 22937 TopSpctps 22959

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-iota 6525 df-fun 6575 df-fv 6581 df-top 22921 df-topon 22938 df-topsp 22960

This theorem is referenced by: tpsprop2d 22966 xmspropd 24504

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