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Mirrors > Home > MPE Home > Th. List > elqtop3 | Structured version Visualization version GIF version |
Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
elqtop3 | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toponuni 21516 | . . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
2 | eqimss 4022 | . . . 4 ⊢ (𝑋 = ∪ 𝐽 → 𝑋 ⊆ ∪ 𝐽) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ⊆ ∪ 𝐽) |
4 | 3 | adantr 483 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → 𝑋 ⊆ ∪ 𝐽) |
5 | eqid 2821 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
6 | 5 | elqtop 22299 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌 ∧ 𝑋 ⊆ ∪ 𝐽) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) |
7 | 4, 6 | mpd3an3 1458 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹:𝑋–onto→𝑌) → (𝐴 ∈ (𝐽 qTop 𝐹) ↔ (𝐴 ⊆ 𝑌 ∧ (◡𝐹 “ 𝐴) ∈ 𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∪ cuni 4831 ◡ccnv 5548 “ cima 5552 –onto→wfo 6347 ‘cfv 6349 (class class class)co 7150 qTop cqtop 16770 TopOnctopon 21512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-qtop 16774 df-topon 21513 |
This theorem is referenced by: qtopid 22307 idqtop 22308 tgqtop 22314 qtopcld 22315 qtopcn 22316 qtopss 22317 qtoprest 22319 qtopomap 22320 kqopn 22336 qtopf1 22418 qustgpopn 22722 |
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