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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 472 | . 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 20536 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
5 | fnex 6386 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐹 ∈ V) | |
6 | 2, 4, 5 | syl2anr 494 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ V) |
7 | fnfun 5902 | . . 3 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹) | |
8 | 7 | adantl 481 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → Fun 𝐹) |
9 | qtoptop2 21312 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ V ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top) | |
10 | 1, 6, 8, 9 | syl3anc 1318 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cuni 4372 Fun wfun 5798 Fn wfn 5799 (class class class)co 6549 qTop cqtop 15986 Topctop 20517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-qtop 15990 df-top 20521 |
This theorem is referenced by: qtoptopon 21317 qtopkgen 21323 qtopt1 29230 qtophaus 29231 |
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