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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 22386 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 4312 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | hmeocn 22370 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾)) | |
4 | cntop1 21850 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ∈ Top) |
6 | cntop2 21851 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
7 | 3, 6 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top) |
8 | eqid 2823 | . . . . . 6 ⊢ (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)) = (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)) | |
9 | 8 | hmeoimaf1o 22380 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)):𝐽–1-1-onto→𝐾) |
10 | f1oen2g 8528 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)):𝐽–1-1-onto→𝐾) → 𝐽 ≈ 𝐾) | |
11 | 5, 7, 9, 10 | syl3anc 1367 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ≈ 𝐾) |
12 | 11 | exlimiv 1931 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ≈ 𝐾) |
13 | 2, 12 | sylbi 219 | . 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → 𝐽 ≈ 𝐾) |
14 | 1, 13 | sylbi 219 | 1 ⊢ (𝐽 ≃ 𝐾 → 𝐽 ≈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 class class class wbr 5068 ↦ cmpt 5148 “ cima 5560 –1-1-onto→wf1o 6356 (class class class)co 7158 ≈ cen 8508 Topctop 21503 Cn ccn 21834 Homeochmeo 22363 ≃ chmph 22364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-1o 8104 df-map 8410 df-en 8512 df-top 21504 df-topon 21521 df-cn 21837 df-hmeo 22365 df-hmph 22366 |
This theorem is referenced by: hmph0 22405 hmphindis 22407 |
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