Theorem hmphen2 23967

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 23944	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4306	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		cmphaushmeo.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
4		cmphaushmeo.2	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
5	3, 4	hmeof1o 23932	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:𝑋–1-1-onto→𝑌)
6		f1oen3g 8961	. . . . 5 ⊢ ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑓:𝑋–1-1-onto→𝑌) → 𝑋 ≈ 𝑌)
7	5, 6	mpdan 699	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌)
8	7	exlimiv 1959	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝑋 ≈ 𝑌)
9	2, 8	sylbi 220	. 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → 𝑋 ≈ 𝑌)
10	1, 9	sylbi 220	1 ⊢ (𝐽 ≃ 𝐾 → 𝑋 ≈ 𝑌)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∃wex 1808   ∈ wcel 2142   ≠ wne 2957  ∅c0 4285  ∪ cuni 4871   class class class wbr 5108  –1-1-onto→wf1o 6535  (class class class)co 7412   ≈ cen 8938  Homeochmeo 23921   ≃ chmph 23922

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-1o 8451  df-map 8824  df-en 8942  df-top 23062  df-topon 23079  df-cn 23395  df-hmeo 23923  df-hmph 23924

This theorem is used by: (None)