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Mirrors > Home > MPE Home > Th. List > conhmph | Structured version Visualization version GIF version |
Description: Connectedness is a topological property. (Contributed by Jeff Hankins, 3-Jul-2009.) |
Ref | Expression |
---|---|
conhmph | ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Con → 𝐾 ∈ Con)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmph 21389 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 3890 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | eqid 2610 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | eqid 2610 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | hmeof1o 21377 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾) |
6 | f1ofo 6057 | . . . . . 6 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝑓:∪ 𝐽–onto→∪ 𝐾) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽–onto→∪ 𝐾) |
8 | hmeocn 21373 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾)) | |
9 | 4 | cnconn 21035 | . . . . . . 7 ⊢ ((𝐽 ∈ Con ∧ 𝑓:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Con) |
10 | 9 | 3expb 1258 | . . . . . 6 ⊢ ((𝐽 ∈ Con ∧ (𝑓:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾))) → 𝐾 ∈ Con) |
11 | 10 | expcom 450 | . . . . 5 ⊢ ((𝑓:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝐽 ∈ Con → 𝐾 ∈ Con)) |
12 | 7, 8, 11 | syl2anc 691 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Con → 𝐾 ∈ Con)) |
13 | 12 | exlimiv 1845 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Con → 𝐾 ∈ Con)) |
14 | 2, 13 | sylbi 206 | . 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Con → 𝐾 ∈ Con)) |
15 | 1, 14 | sylbi 206 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Con → 𝐾 ∈ Con)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∅c0 3874 ∪ cuni 4372 class class class wbr 4583 –onto→wfo 5802 –1-1-onto→wf1o 5803 (class class class)co 6549 Cn ccn 20838 Conccon 21024 Homeochmeo 21366 ≃ chmph 21367 |
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-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-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-suc 5646 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-1st 7059 df-2nd 7060 df-1o 7447 df-map 7746 df-top 20521 df-topon 20523 df-cld 20633 df-cn 20841 df-con 21025 df-hmeo 21368 df-hmph 21369 |
This theorem is referenced by: xrcon 22556 |
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