Theorem hmphtop1 23907

Description: The relation "being homeomorphic to" implies the operands are topologies. (Contributed by FL, 23-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
hmphtop1	⊢ (𝐽 ≃ 𝐾 → 𝐽 ∈ Top)

Proof of Theorem hmphtop1

Step	Hyp	Ref	Expression
1		hmphtop 23906	. 2 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top))
2	1	simpld 499	1 ⊢ (𝐽 ≃ 𝐾 → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2149   class class class wbr 5113  Topctop 23021   ≃ chmph 23882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5273  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7988  df-2nd 7989  df-hmeo 23883  df-hmph 23884

This theorem is referenced by:  hmpher  23912  hmph0  23923  hmphindis  23925  kqhmph  23947