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| 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 23784 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
| 2 | n0 4353 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
| 3 | hmeocn 23768 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾)) | |
| 4 | cntop1 23248 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
| 5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ∈ Top) | 
| 6 | cntop2 23249 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
| 7 | 3, 6 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top) | 
| 8 | eqid 2737 | . . . . . 6 ⊢ (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)) = (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)) | |
| 9 | 8 | hmeoimaf1o 23778 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)):𝐽–1-1-onto→𝐾) | 
| 10 | f1oen2g 9009 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)):𝐽–1-1-onto→𝐾) → 𝐽 ≈ 𝐾) | |
| 11 | 5, 7, 9, 10 | syl3anc 1373 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ≈ 𝐾) | 
| 12 | 11 | exlimiv 1930 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ≈ 𝐾) | 
| 13 | 2, 12 | sylbi 217 | . 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → 𝐽 ≈ 𝐾) | 
| 14 | 1, 13 | sylbi 217 | 1 ⊢ (𝐽 ≃ 𝐾 → 𝐽 ≈ 𝐾) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 class class class wbr 5143 ↦ cmpt 5225 “ cima 5688 –1-1-onto→wf1o 6560 (class class class)co 7431 ≈ cen 8982 Topctop 22899 Cn ccn 23232 Homeochmeo 23761 ≃ chmph 23762 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-1o 8506 df-map 8868 df-en 8986 df-top 22900 df-topon 22917 df-cn 23235 df-hmeo 23763 df-hmph 23764 | 
| This theorem is referenced by: hmph0 23803 hmphindis 23805 | 
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