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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 22912 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
| 5 | fnex 7237 | . . 3 ⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝐹 ∈ V) | |
| 6 | 2, 4, 5 | syl2anr 597 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ V) |
| 7 | fnfun 6668 | . . 3 ⊢ (𝐹 Fn 𝑋 → Fun 𝐹) | |
| 8 | 7 | adantl 481 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → Fun 𝐹) |
| 9 | qtoptop2 23707 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ V ∧ Fun 𝐹) → (𝐽 qTop 𝐹) ∈ Top) | |
| 10 | 1, 6, 8, 9 | syl3anc 1373 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Top) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cuni 4907 Fun wfun 6555 Fn wfn 6556 (class class class)co 7431 qTop cqtop 17548 Topctop 22899 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-qtop 17552 df-top 22900 |
| This theorem is referenced by: qtoptopon 23712 qtopkgen 23718 qtopt1 33834 qtophaus 33835 |
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