Theorem hmphen2 23723

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 23700	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4350	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		cmphaushmeo.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
4		cmphaushmeo.2	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
5	3, 4	hmeof1o 23688	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋–1-1-onto→𝑌)
6		f1oen3g 8993	. . . . 5 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋–1-1-onto→𝑌) → 𝑋 ≈ 𝑌)
7	5, 6	mpdan 685	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌)
8	7	exlimiv 1925	. . 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 1533  ∃wex 1773   ∈ wcel 2098   ≠ wne 2937  ∅c0 4326  ∪ cuni 4912   class class class wbr 5152  –1-1-onto→wf1o 6552  (class class class)co 7426   ≈ cen 8967  Homeochmeo 23677   ≃ chmph 23678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-1o 8493  df-map 8853  df-en 8971  df-top 22816  df-topon 22833  df-cn 23151  df-hmeo 23679  df-hmph 23680

This theorem is referenced by: (None)