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Theorem topnpropd 17481

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 7449	. 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾_t (Base‘𝐾)) = ((TopSet‘𝐿) ↾_t (Base‘𝐿)))
4		eqid 2737	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2737	. . 3 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
6	4, 5	topnval 17479	. 2 ⊢ ((TopSet‘𝐾) ↾_t (Base‘𝐾)) = (TopOpen‘𝐾)
7		eqid 2737	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
8		eqid 2737	. . 3 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿)
9	7, 8	topnval 17479	. 2 ⊢ ((TopSet‘𝐿) ↾_t (Base‘𝐿)) = (TopOpen‘𝐿)
10	3, 6, 9	3eqtr3g 2800	1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 TopSetcts 17303 ↾_t crest 17465 TopOpenctopn 17466

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-rest 17467 df-topn 17468

This theorem is referenced by: sratopn 21190 tpsprop2d 22945 nrgtrg 24711 zhmnrg 33966