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Mirrors > Home > MPE Home > Th. List > hmphen | Structured version Visualization version GIF version |
Description: Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
hmphen | ⊢ (𝐽 ≃ 𝐾 → 𝐽 ≈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmph 22381 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 4260 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | hmeocn 22365 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾)) | |
4 | cntop1 21845 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ∈ Top) |
6 | cntop2 21846 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
7 | 3, 6 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top) |
8 | eqid 2798 | . . . . . 6 ⊢ (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)) = (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)) | |
9 | 8 | hmeoimaf1o 22375 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)):𝐽–1-1-onto→𝐾) |
10 | f1oen2g 8509 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)):𝐽–1-1-onto→𝐾) → 𝐽 ≈ 𝐾) | |
11 | 5, 7, 9, 10 | syl3anc 1368 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ≈ 𝐾) |
12 | 11 | exlimiv 1931 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ≈ 𝐾) |
13 | 2, 12 | sylbi 220 | . 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → 𝐽 ≈ 𝐾) |
14 | 1, 13 | sylbi 220 | 1 ⊢ (𝐽 ≃ 𝐾 → 𝐽 ≈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 class class class wbr 5030 ↦ cmpt 5110 “ cima 5522 –1-1-onto→wf1o 6323 (class class class)co 7135 ≈ cen 8489 Topctop 21498 Cn ccn 21829 Homeochmeo 22358 ≃ chmph 22359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-1o 8085 df-map 8391 df-en 8493 df-top 21499 df-topon 21516 df-cn 21832 df-hmeo 22360 df-hmph 22361 |
This theorem is referenced by: hmph0 22400 hmphindis 22402 |
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