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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 21579 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 3931 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | cmphaushmeo.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
4 | cmphaushmeo.2 | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
5 | 3, 4 | hmeof1o 21567 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋–1-1-onto→𝑌) |
6 | f1oen3g 7971 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋–1-1-onto→𝑌) → 𝑋 ≈ 𝑌) | |
7 | 5, 6 | mpdan 702 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌) |
8 | 7 | exlimiv 1858 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌) |
9 | 2, 8 | sylbi 207 | . 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → 𝑋 ≈ 𝑌) |
10 | 1, 9 | sylbi 207 | 1 ⊢ (𝐽 ≃ 𝐾 → 𝑋 ≈ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 ∪ cuni 4436 class class class wbr 4653 –1-1-onto→wf1o 5887 (class class class)co 6650 ≈ cen 7952 Homeochmeo 21556 ≃ chmph 21557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-1o 7560 df-map 7859 df-en 7956 df-top 20699 df-topon 20716 df-cn 21031 df-hmeo 21558 df-hmph 21559 |
This theorem is referenced by: (None) |
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