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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 485 | . 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 21517 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
5 | fnex 6983 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐹 ∈ V) | |
6 | 2, 4, 5 | syl2anr 598 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ V) |
7 | fnfun 6456 | . . 3 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹) | |
8 | 7 | adantl 484 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → Fun 𝐹) |
9 | qtoptop2 22310 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ V ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top) | |
10 | 1, 6, 8, 9 | syl3anc 1367 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∪ cuni 4841 Fun wfun 6352 Fn wfn 6353 (class class class)co 7159 qTop cqtop 16779 Topctop 21504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-qtop 16783 df-top 21505 |
This theorem is referenced by: qtoptopon 22315 qtopkgen 22321 qtopt1 31103 qtophaus 31104 |
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