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Mirrors > Home > MPE Home > Th. List > qtopconn | Structured version Visualization version GIF version |
Description: A quotient of a connected space is connected. (Contributed by Mario Carneiro, 24-Mar-2015.) |
Ref | Expression |
---|---|
qtopcmp.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
qtopconn | ⊢ ((𝐽 ∈ Conn ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Conn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qtopcmp.1 | . 2 ⊢ 𝑋 = ∪ 𝐽 | |
2 | conntop 21741 | . 2 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top) | |
3 | eqid 2772 | . . 3 ⊢ ∪ (𝐽 qTop 𝐹) = ∪ (𝐽 qTop 𝐹) | |
4 | 3 | cnconn 21746 | . 2 ⊢ ((𝐽 ∈ Conn ∧ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ Conn) |
5 | 1, 2, 4 | qtopcmplem 22031 | 1 ⊢ ((𝐽 ∈ Conn ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ Conn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∪ cuni 4708 Fn wfn 6180 (class class class)co 6974 qTop cqtop 16630 Conncconn 21735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-map 8206 df-qtop 16634 df-top 21218 df-topon 21235 df-cld 21343 df-cn 21551 df-conn 21736 |
This theorem is referenced by: (None) |
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