![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > qtopuni | Structured version Visualization version GIF version |
Description: The base set of the quotient topology. (Contributed by Mario Carneiro, 23-Mar-2015.) |
Ref | Expression |
---|---|
qtoptop.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
qtopuni | ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssidd 3971 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 ⊆ 𝑌) | |
2 | fof 6760 | . . . . . . 7 ⊢ (𝐹:𝑋–onto→𝑌 → 𝐹:𝑋⟶𝑌) | |
3 | 2 | adantl 483 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝐹:𝑋⟶𝑌) |
4 | fimacnv 6694 | . . . . . 6 ⊢ (𝐹:𝑋⟶𝑌 → (◡𝐹 “ 𝑌) = 𝑋) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → (◡𝐹 “ 𝑌) = 𝑋) |
6 | qtoptop.1 | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
7 | 6 | topopn 22278 | . . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
8 | 7 | adantr 482 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑋 ∈ 𝐽) |
9 | 5, 8 | eqeltrd 2834 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → (◡𝐹 “ 𝑌) ∈ 𝐽) |
10 | 6 | elqtop2 23075 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → (𝑌 ∈ (𝐽 qTop 𝐹) ↔ (𝑌 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑌) ∈ 𝐽))) |
11 | 1, 9, 10 | mpbir2and 712 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 ∈ (𝐽 qTop 𝐹)) |
12 | elssuni 4902 | . . 3 ⊢ (𝑌 ∈ (𝐽 qTop 𝐹) → 𝑌 ⊆ ∪ (𝐽 qTop 𝐹)) | |
13 | 11, 12 | syl 17 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 ⊆ ∪ (𝐽 qTop 𝐹)) |
14 | 6 | elqtop2 23075 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽))) |
15 | simpl 484 | . . . . . 6 ⊢ ((𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) → 𝑥 ⊆ 𝑌) | |
16 | velpw 4569 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝑌 ↔ 𝑥 ⊆ 𝑌) | |
17 | 15, 16 | sylibr 233 | . . . . 5 ⊢ ((𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) → 𝑥 ∈ 𝒫 𝑌) |
18 | 14, 17 | syl6bi 253 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) → 𝑥 ∈ 𝒫 𝑌)) |
19 | 18 | ssrdv 3954 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ⊆ 𝒫 𝑌) |
20 | sspwuni 5064 | . . 3 ⊢ ((𝐽 qTop 𝐹) ⊆ 𝒫 𝑌 ↔ ∪ (𝐽 qTop 𝐹) ⊆ 𝑌) | |
21 | 19, 20 | sylib 217 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → ∪ (𝐽 qTop 𝐹) ⊆ 𝑌) |
22 | 13, 21 | eqssd 3965 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3914 𝒫 cpw 4564 ∪ cuni 4869 ◡ccnv 5636 “ cima 5640 ⟶wf 6496 –onto→wfo 6498 (class class class)co 7361 qTop cqtop 17393 Topctop 22265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-qtop 17397 df-top 22266 |
This theorem is referenced by: qtoptopon 23078 qtopcmplem 23081 qtopkgen 23084 qtopt1 32480 qtophaus 32481 circtopn 32482 |
Copyright terms: Public domain | W3C validator |