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Mirrors > Home > MPE Home > Th. List > qtoptopon | Structured version Visualization version GIF version |
Description: The base set of the quotient topology. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
qtoptopon | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponuni 20542 | . . . . . 6 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
2 | foeq2 6025 | . . . . . 6 ⊢ (𝑋 = ∪ 𝐽 → (𝐹:𝑋–onto→𝑌 ↔ 𝐹:∪ 𝐽–onto→𝑌)) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐹:𝑋–onto→𝑌 ↔ 𝐹:∪ 𝐽–onto→𝑌)) |
4 | 3 | biimpa 500 | . . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → 𝐹:∪ 𝐽–onto→𝑌) |
5 | fofn 6030 | . . . 4 ⊢ (𝐹:∪ 𝐽–onto→𝑌 → 𝐹 Fn ∪ 𝐽) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → 𝐹 Fn ∪ 𝐽) |
7 | topontop 20541 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | |
8 | eqid 2610 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
9 | 8 | qtoptop 21313 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹 Fn ∪ 𝐽) → (𝐽 qTop 𝐹) ∈ Top) |
10 | 7, 9 | sylan 487 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn ∪ 𝐽) → (𝐽 qTop 𝐹) ∈ Top) |
11 | 6, 10 | syldan 486 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ Top) |
12 | 8 | qtopuni 21315 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐹:∪ 𝐽–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
13 | 7, 12 | sylan 487 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:∪ 𝐽–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
14 | 4, 13 | syldan 486 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → 𝑌 = ∪ (𝐽 qTop 𝐹)) |
15 | istopon 20540 | . 2 ⊢ ((𝐽 qTop 𝐹) ∈ (TopOn‘𝑌) ↔ ((𝐽 qTop 𝐹) ∈ Top ∧ 𝑌 = ∪ (𝐽 qTop 𝐹))) | |
16 | 11, 14, 15 | sylanbrc 695 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐽 qTop 𝐹) ∈ (TopOn‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∪ cuni 4372 Fn wfn 5799 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 qTop cqtop 15986 Topctop 20517 TopOnctopon 20518 |
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 df-topon 20523 |
This theorem is referenced by: qtopid 21318 qtopcld 21326 qtopcn 21327 qtopeu 21329 qtoprest 21330 imastps 21334 kqtopon 21340 qtopf1 21429 qtophmeo 21430 qustgplem 21734 qtophaus 29231 |
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