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| Description: Conditions that determine the equality of two generated topologies. (Contributed by NM, 8-May-2007.) (Revised by Mario Carneiro, 2-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| 2basgen | ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → (topGen‘𝐵) = (topGen‘𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fvex 6918 | . . . 4 ⊢ (topGen‘𝐵) ∈ V | |
| 2 | 1 | ssex 5320 | . . 3 ⊢ (𝐶 ⊆ (topGen‘𝐵) → 𝐶 ∈ V) | 
| 3 | simpl 482 | . . 3 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → 𝐵 ⊆ 𝐶) | |
| 4 | tgss 22976 | . . 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 5322 | . . . . 5 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ∈ V) → 𝐵 ∈ V) | |
| 8 | 2, 7 | sylan2 593 | . . . 4 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘𝐵)) → 𝐵 ∈ V) | 
| 9 | tgss3 22994 | . . . 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 4000 | 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 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 ‘cfv 6560 topGenctg 17483 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-topgen 17489 | 
| This theorem is referenced by: leordtval2 23221 2ndcsb 23458 txbasval 23615 prdsxmslem2 24543 tgioo 24818 tgqioo 24822 | 
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