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Theorem topnpropd 17417

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 7434	. 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾_t (Base‘𝐾)) = ((TopSet‘𝐿) ↾_t (Base‘𝐿)))
4		eqid 2725	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2725	. . 3 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
6	4, 5	topnval 17415	. 2 ⊢ ((TopSet‘𝐾) ↾_t (Base‘𝐾)) = (TopOpen‘𝐾)
7		eqid 2725	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
8		eqid 2725	. . 3 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿)
9	7, 8	topnval 17415	. 2 ⊢ ((TopSet‘𝐿) ↾_t (Base‘𝐿)) = (TopOpen‘𝐿)
10	3, 6, 9	3eqtr3g 2788	1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ‘cfv 6543 (class class class)co 7416 Basecbs 17179 TopSetcts 17238 ↾_t crest 17401 TopOpenctopn 17402

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 df-rest 17403 df-topn 17404

This theorem is referenced by: sratopn 21080 tpsprop2d 22859 nrgtrg 24625 zhmnrg 33625