topnpropd - Metamath Proof Explorer

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Theorem topnpropd 17607

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 7438	. 2 ⊢ (𝜑 → ((TopSet‘𝐾) ↾_t (Base‘𝐾)) = ((TopSet‘𝐿) ↾_t (Base‘𝐿)))
4		eqid 2760	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2760	. . 3 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
6	4, 5	topnval 17605	. 2 ⊢ ((TopSet‘𝐾) ↾_t (Base‘𝐾)) = (TopOpen‘𝐾)
7		eqid 2760	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
8		eqid 2760	. . 3 ⊢ (TopSet‘𝐿) = (TopSet‘𝐿)
9	7, 8	topnval 17605	. 2 ⊢ ((TopSet‘𝐿) ↾_t (Base‘𝐿)) = (TopOpen‘𝐿)
10	3, 6, 9	3eqtr3g 2818	1 ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1569 ‘cfv 6538 (class class class)co 7420 Basecbs 17387 TopSetcts 17434 ↾_t crest 17591 TopOpenctopn 17592

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-rest 17593 df-topn 17594

This theorem is used by: sratopn 21459 tpsprop2d 23257 nrgtrg 25009 zhmnrg 34597

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