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Theorem kgencmp 21158
 Description: The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.)
Assertion
Ref Expression
kgencmp ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) = ((𝑘Gen‘𝐽) ↾t 𝐾))

Proof of Theorem kgencmp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 kgenftop 21153 . . . 4 (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top)
21adantr 480 . . 3 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝑘Gen‘𝐽) ∈ Top)
3 kgenss 21156 . . . 4 (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
43adantr 480 . . 3 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → 𝐽 ⊆ (𝑘Gen‘𝐽))
5 ssrest 20790 . . 3 (((𝑘Gen‘𝐽) ∈ Top ∧ 𝐽 ⊆ (𝑘Gen‘𝐽)) → (𝐽t 𝐾) ⊆ ((𝑘Gen‘𝐽) ↾t 𝐾))
62, 4, 5syl2anc 691 . 2 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ⊆ ((𝑘Gen‘𝐽) ↾t 𝐾))
7 cmptop 21008 . . . . . 6 ((𝐽t 𝐾) ∈ Comp → (𝐽t 𝐾) ∈ Top)
87adantl 481 . . . . 5 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Top)
9 restrcl 20771 . . . . . 6 ((𝐽t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V))
109simprd 478 . . . . 5 ((𝐽t 𝐾) ∈ Top → 𝐾 ∈ V)
118, 10syl 17 . . . 4 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → 𝐾 ∈ V)
12 restval 15910 . . . 4 (((𝑘Gen‘𝐽) ∈ Top ∧ 𝐾 ∈ V) → ((𝑘Gen‘𝐽) ↾t 𝐾) = ran (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)))
132, 11, 12syl2anc 691 . . 3 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → ((𝑘Gen‘𝐽) ↾t 𝐾) = ran (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)))
14 simpr 476 . . . . . 6 (((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ∈ (𝑘Gen‘𝐽))
15 simplr 788 . . . . . 6 (((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝐽t 𝐾) ∈ Comp)
16 kgeni 21150 . . . . . 6 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝑥𝐾) ∈ (𝐽t 𝐾))
1714, 15, 16syl2anc 691 . . . . 5 (((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥𝐾) ∈ (𝐽t 𝐾))
18 eqid 2610 . . . . 5 (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)) = (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾))
1917, 18fmptd 6292 . . . 4 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)):(𝑘Gen‘𝐽)⟶(𝐽t 𝐾))
20 frn 5966 . . . 4 ((𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)):(𝑘Gen‘𝐽)⟶(𝐽t 𝐾) → ran (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)) ⊆ (𝐽t 𝐾))
2119, 20syl 17 . . 3 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → ran (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)) ⊆ (𝐽t 𝐾))
2213, 21eqsstrd 3602 . 2 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → ((𝑘Gen‘𝐽) ↾t 𝐾) ⊆ (𝐽t 𝐾))
236, 22eqssd 3585 1 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) = ((𝑘Gen‘𝐽) ↾t 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  Compccmp 20999  𝑘Genckgen 21146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-kgen 21147 This theorem is referenced by:  kgencmp2  21159  kgenidm  21160
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