tpspropd - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tpspropd		Structured version Visualization version GIF version

Theorem tpspropd 22882

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 6838	. . 3 ⊢ (𝜑 → (TopOn‘(Base‘𝐾)) = (TopOn‘(Base‘𝐿)))
4	1, 3	eleq12d 2830	. 2 ⊢ (𝜑 → ((TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)) ↔ (TopOpen‘𝐿) ∈ (TopOn‘(Base‘𝐿))))
5		eqid 2736	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
6		eqid 2736	. . 3 ⊢ (TopOpen‘𝐾) = (TopOpen‘𝐾)
7	5, 6	istps 22878	. 2 ⊢ (𝐾 ∈ TopSp ↔ (TopOpen‘𝐾) ∈ (TopOn‘(Base‘𝐾)))
8		eqid 2736	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
9		eqid 2736	. . 3 ⊢ (TopOpen‘𝐿) = (TopOpen‘𝐿)
10	8, 9	istps 22878	. 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 6492 Basecbs 17136 TopOpenctopn 17341 TopOnctopon 22854 TopSpctps 22876

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 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680

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 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-iota 6448 df-fun 6494 df-fv 6500 df-top 22838 df-topon 22855 df-topsp 22877

This theorem is referenced by: tpsprop2d 22883 xmspropd 24417

W3C validator