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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 23800 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 4359 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | cmphaushmeo.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
4 | cmphaushmeo.2 | . . . . . 6 ⊢ 𝑌 = ∪ 𝐾 | |
5 | 3, 4 | hmeof1o 23788 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋–1-1-onto→𝑌) |
6 | f1oen3g 9006 | . . . . 5 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋–1-1-onto→𝑌) → 𝑋 ≈ 𝑌) | |
7 | 5, 6 | mpdan 687 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌) |
8 | 7 | exlimiv 1928 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌) |
9 | 2, 8 | sylbi 217 | . 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → 𝑋 ≈ 𝑌) |
10 | 1, 9 | sylbi 217 | 1 ⊢ (𝐽 ≃ 𝐾 → 𝑋 ≈ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 ∪ cuni 4912 class class class wbr 5148 –1-1-onto→wf1o 6562 (class class class)co 7431 ≈ cen 8981 Homeochmeo 23777 ≃ chmph 23778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-1o 8505 df-map 8867 df-en 8985 df-top 22916 df-topon 22933 df-cn 23251 df-hmeo 23779 df-hmph 23780 |
This theorem is referenced by: (None) |
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