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Theorem 2basgen 22906

Description: Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.)

**Assertion**
Ref	Expression
2basgen	⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))

**Proof of Theorem 2basgen**
Step	Hyp	Ref	Expression
1		fvex 6841	. . . 4 ⊢ (topGen‘𝐵) ∈ V
2	1	ssex 5261	. . 3 ⊢ (𝐶 ⊆ (topGen‘𝐵) → 𝐶 ∈ V)
3		simpl 482	. . 3 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → 𝐵 ⊆ 𝐶)
4		tgss 22884	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐵 ⊆ 𝐶) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
5	2, 3, 4	syl2an2 686	. 2 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) ⊆ (topGen‘𝐶))
6		simpr 484	. . 3 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → 𝐶 ⊆ (topGen‘𝐵))
7		ssexg 5263	. . . . 5 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ∈ V) → 𝐵 ∈ V)
8	2, 7	sylan2 593	. . . 4 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V)
9		tgss3 22902	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐵 ∈ V) → ((topGen‘𝐶) ⊆ (topGen‘𝐵) ↔ 𝐶 ⊆ (topGen‘𝐵)))
10	2, 8, 9	syl2an2 686	. . 3 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → ((topGen‘𝐶) ⊆ (topGen‘𝐵) ↔ 𝐶 ⊆ (topGen‘𝐵)))
11	6, 10	mpbird 257	. 2 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐶) ⊆ (topGen‘𝐵))
12	5, 11	eqssd 3948	1 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3437 ⊆ wss 3898 ‘cfv 6486 topGenctg 17343

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 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-iota 6442 df-fun 6488 df-fv 6494 df-topgen 17349

This theorem is referenced by: leordtval2 23128 2ndcsb 23365 txbasval 23522 prdsxmslem2 24445 tgioo 24712 tgqioo 24716

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