topnpropd - Metamath Proof Explorer

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Theorem topnpropd 17340

Description: The topology extractor function depends only on the base and topology components. (Contributed by NM, 18-Jul-2006.)

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

**Assertion**
Ref	Expression
topnpropd	⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

**Proof of Theorem topnpropd**
Step	Hyp	Ref	Expression
1		topnpropd.2	. . 3 ⊢ (𝜑 → (TopSet‘𝐾) = (TopSet‘𝐿))
2		topnpropd.1	. . 3 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
3	1, 2	oveq12d 7364	. 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾_t (Base‘𝐾)) = ((TopSet‘𝐿) ↾_t (Base‘𝐿)))
4		eqid 2731	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2731	. . 3 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
6	4, 5	topnval 17338	. 2 ⊢ ((TopSet‘𝐾) ↾_t (Base‘𝐾)) = (TopOpen‘𝐾)
7		eqid 2731	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
8		eqid 2731	. . 3 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿)
9	7, 8	topnval 17338	. 2 ⊢ ((TopSet‘𝐿) ↾_t (Base‘𝐿)) = (TopOpen‘𝐿)
10	3, 6, 9	3eqtr3g 2789	1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 TopSetcts 17167 ↾_t crest 17324 TopOpenctopn 17325

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-rest 17326 df-topn 17327

This theorem is referenced by: sratopn 21118 tpsprop2d 22854 nrgtrg 24605 zhmnrg 33978