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Mirrors > Home > MPE Home > Th. List > qtoptop | Structured version Visualization version GIF version |
Description: The quotient topology is a topology. (Contributed by Mario Carneiro, 23-Mar-2015.) |
Ref | Expression |
---|---|
qtoptop.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
qtoptop | ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → 𝐽 ∈ Top) | |
2 | id 22 | . . 3 ⊢ (𝐹 Fn 𝑋 → 𝐹 Fn 𝑋) | |
3 | qtoptop.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | topopn 22730 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
5 | fnex 7210 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐹 ∈ V) | |
6 | 2, 4, 5 | syl2anr 596 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ V) |
7 | fnfun 6639 | . . 3 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹) | |
8 | 7 | adantl 481 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → Fun 𝐹) |
9 | qtoptop2 23525 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ V ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top) | |
10 | 1, 6, 8, 9 | syl3anc 1368 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ∪ cuni 4899 Fun wfun 6527 Fn wfn 6528 (class class class)co 7401 qTop cqtop 17448 Topctop 22717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-qtop 17452 df-top 22718 |
This theorem is referenced by: qtoptopon 23530 qtopkgen 23536 qtopt1 33304 qtophaus 33305 |
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