Theorem hmphen 21588

Description: Homeomorphisms preserve the cardinality of the topologies. (Contributed by FL, 1-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
hmphen	⊢ (𝐽 ≃ 𝐾 → 𝐽 ≈ 𝐾)

Proof of Theorem hmphen

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmph 21579	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 3931	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		hmeocn 21563	. . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
4		cntop1 21044	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
5	3, 4	syl 17	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ∈ Top)
6		cntop2 21045	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
7	3, 6	syl 17	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ∈ Top)
8		eqid 2622	. . . . . 6 ⊢ (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)) = (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥))
9	8	hmeoimaf1o 21573	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)):𝐽–1-1-onto→𝐾)
10		f1oen2g 7972	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ (𝑥 ∈ 𝐽 ↦ (𝑓 “ 𝑥)):𝐽–1-1-onto→𝐾) → 𝐽 ≈ 𝐾)
11	5, 7, 9, 10	syl3anc 1326	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ≈ 𝐾)
12	11	exlimiv 1858	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝐽 ≈ 𝐾)
13	2, 12	sylbi 207	. 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → 𝐽 ≈ 𝐾)
14	1, 13	sylbi 207	1 ⊢ (𝐽 ≃ 𝐾 → 𝐽 ≈ 𝐾)

Syntax hints:   → wi 4  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∅c0 3915   class class class wbr 4653   ↦ cmpt 4729   “ cima 5117  –1-1-onto→wf1o 5887  (class class class)co 6650   ≈ cen 7952  Topctop 20698   Cn ccn 21028  Homeochmeo 21556   ≃ chmph 21557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-1o 7560  df-map 7859  df-en 7956  df-top 20699  df-topon 20716  df-cn 21031  df-hmeo 21558  df-hmph 21559

This theorem is referenced by:  hmph0  21598  hmphindis  21600