Theorem hmphen2 23742

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 23719	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4333	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		cmphaushmeo.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
4		cmphaushmeo.2	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
5	3, 4	hmeof1o 23707	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋–1-1-onto→𝑌)
6		f1oen3g 8986	. . . . 5 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋–1-1-onto→𝑌) → 𝑋 ≈ 𝑌)
7	5, 6	mpdan 687	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌)
8	7	exlimiv 1930	. . 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)

Syntax hints:   → wi 4   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∅c0 4313  ∪ cuni 4888   class class class wbr 5124  –1-1-onto→wf1o 6535  (class class class)co 7410   ≈ cen 8961  Homeochmeo 23696   ≃ chmph 23697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-1o 8485  df-map 8847  df-en 8965  df-top 22837  df-topon 22854  df-cn 23170  df-hmeo 23698  df-hmph 23699

This theorem is referenced by: (None)