Theorem cmphmph 24137

Description: Compactness is a topological property-that is, for any two homeomorphic topologies, either both are compact or neither is. (Contributed by Jeff Hankins, 30-Jun-2009.) (Revised by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
cmphmph	⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Comp → 𝐾 ∈ Comp))

Proof of Theorem cmphmph

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmph 24125	. 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2		n0 4303	. . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3		eqid 2763	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2763	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	hmeof1o 24113	. . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽⟶1-1onto∪ 𝐾)
6		f1ofo 6854	. . . . . 6 ⊢ (𝑓:∪ 𝐽⟶1-1onto∪ 𝐾 → 𝑓:∪ 𝐽⟶onto∪ 𝐾)
7	5, 6	syl 18	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽⟶onto∪ 𝐾)
8		hmeocn 24109	. . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
9	4	cncmp 23740	. . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝑓:∪ 𝐽⟶onto∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)
10	9	3expb 1138	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ (𝑓:∪ 𝐽⟶onto∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾))) → 𝐾 ∈ Comp)
11	10	expcom 419	. . . . 5 ⊢ ((𝑓:∪ 𝐽⟶onto∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝐽 ∈ Comp → 𝐾 ∈ Comp))
12	7, 8, 11	syl2anc 596	. . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Comp → 𝐾 ∈ Comp))
13	12	exlimiv 1965	. . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Comp → 𝐾 ∈ Comp))
14	2, 13	sylbi 220	. 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Comp → 𝐾 ∈ Comp))
15	1, 14	sylbi 220	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Comp → 𝐾 ∈ Comp))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  ∅c0 4281  ∪ cuni 4870   class class class wbr 5106  ⟶ontowfo 6556  ⟶1-1ontowf1o 6557  (class class class)co 7443   Cn ccn 23572  Compccmp 23734  Homeochmeo 24102   ≃ chmph 24103

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-1o 8504  df-map 8877  df-en 9002  df-dom 9003  df-fin 9005  df-top 23242  df-topon 23259  df-cn 23575  df-cmp 23735  df-hmeo 24104  df-hmph 24105

This theorem is used by:  ptcmpfi  24162  xrcmp  25299  reheibor  38965