Theorem hmphen2 22950

Description: Homeomorphisms preserve the cardinality of the underlying sets. (Contributed by FL, 17-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Hypotheses

Ref	Expression
cmphaushmeo.1	⊢ 𝑋 = ∪ 𝐽
cmphaushmeo.2	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
hmphen2	⊢ (𝐽 ≃ 𝐾 → 𝑋 ≈ 𝑌)

Proof of Theorem hmphen2

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmph 22927	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4280	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		cmphaushmeo.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
4		cmphaushmeo.2	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
5	3, 4	hmeof1o 22915	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋–1-1-onto→𝑌)
6		f1oen3g 8754	. . . . 5 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋–1-1-onto→𝑌) → 𝑋 ≈ 𝑌)
7	5, 6	mpdan 684	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌)
8	7	exlimiv 1933	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌)
9	2, 8	sylbi 216	. 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → 𝑋 ≈ 𝑌)
10	1, 9	sylbi 216	1 ⊢ (𝐽 ≃ 𝐾 → 𝑋 ≈ 𝑌)

Syntax hints:   → wi 4   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  ∪ cuni 4839   class class class wbr 5074  –1-1-onto→wf1o 6432  (class class class)co 7275   ≈ cen 8730  Homeochmeo 22904   ≃ chmph 22905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-1o 8297  df-map 8617  df-en 8734  df-top 22043  df-topon 22060  df-cn 22378  df-hmeo 22906  df-hmph 22907

This theorem is referenced by: (None)