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Mirrors > Home > MPE Home > Th. List > hmphen2 | Structured version Visualization version GIF version |
Description: Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
cmphaushmeo.1 | ⊢ 𝑋 = ∪ 𝐽 |
cmphaushmeo.2 | ⊢ 𝑌 = ∪ 𝐾 |
Ref | Expression |
---|---|
hmphen2 | ⊢ (𝐽 ≃ 𝐾 → 𝑋 ≈ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmph 21950 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 4160 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | cmphaushmeo.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
4 | cmphaushmeo.2 | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
5 | 3, 4 | hmeof1o 21938 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋–1-1-onto→𝑌) |
6 | f1oen3g 8238 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋–1-1-onto→𝑌) → 𝑋 ≈ 𝑌) | |
7 | 5, 6 | mpdan 680 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌) |
8 | 7 | exlimiv 2031 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌) |
9 | 2, 8 | sylbi 209 | . 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → 𝑋 ≈ 𝑌) |
10 | 1, 9 | sylbi 209 | 1 ⊢ (𝐽 ≃ 𝐾 → 𝑋 ≈ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∃wex 1880 ∈ wcel 2166 ≠ wne 2999 ∅c0 4144 ∪ cuni 4658 class class class wbr 4873 –1-1-onto→wf1o 6122 (class class class)co 6905 ≈ cen 8219 Homeochmeo 21927 ≃ chmph 21928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-1st 7428 df-2nd 7429 df-1o 7826 df-map 8124 df-en 8223 df-top 21069 df-topon 21086 df-cn 21402 df-hmeo 21929 df-hmph 21930 |
This theorem is referenced by: (None) |
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