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Mirrors > Home > MPE Home > Th. List > sshauslem | Structured version Visualization version GIF version |
Description: Lemma for sshaus 21549 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then a topology finer than one with property 𝐴 also has property 𝐴. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
t1sep.1 | ⊢ 𝑋 = ∪ 𝐽 |
sshauslem.2 | ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top) |
sshauslem.3 | ⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ 𝑋):𝑋–1-1→𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴) |
Ref | Expression |
---|---|
sshauslem | ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1172 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ 𝐴) | |
2 | f1oi 6414 | . . 3 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
3 | f1of1 6376 | . . 3 ⊢ (( I ↾ 𝑋):𝑋–1-1-onto→𝑋 → ( I ↾ 𝑋):𝑋–1-1→𝑋) | |
4 | 2, 3 | mp1i 13 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → ( I ↾ 𝑋):𝑋–1-1→𝑋) |
5 | simp3 1174 | . . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ⊆ 𝐾) | |
6 | simp2 1173 | . . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ (TopOn‘𝑋)) | |
7 | sshauslem.2 | . . . . . 6 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top) | |
8 | 7 | 3ad2ant1 1169 | . . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Top) |
9 | t1sep.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
10 | 9 | toptopon 21091 | . . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
11 | 8, 10 | sylib 210 | . . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ (TopOn‘𝑋)) |
12 | ssidcn 21429 | . . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽) ↔ 𝐽 ⊆ 𝐾)) | |
13 | 6, 11, 12 | syl2anc 581 | . . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽) ↔ 𝐽 ⊆ 𝐾)) |
14 | 5, 13 | mpbird 249 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) |
15 | sshauslem.3 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ 𝑋):𝑋–1-1→𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴) | |
16 | 1, 4, 14, 15 | syl3anc 1496 | 1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ⊆ wss 3797 ∪ cuni 4657 I cid 5248 ↾ cres 5343 –1-1→wf1 6119 –1-1-onto→wf1o 6121 ‘cfv 6122 (class class class)co 6904 Topctop 21067 TopOnctopon 21084 Cn ccn 21398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-map 8123 df-top 21068 df-topon 21085 df-cn 21401 |
This theorem is referenced by: sst0 21547 sst1 21548 sshaus 21549 |
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