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Theorem kgencmp 22146
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 22141 . . . 4 (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top)
21adantr 484 . . 3 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝑘Gen‘𝐽) ∈ Top)
3 kgenss 22144 . . . 4 (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽))
43adantr 484 . . 3 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → 𝐽 ⊆ (𝑘Gen‘𝐽))
5 ssrest 21777 . . 3 (((𝑘Gen‘𝐽) ∈ Top ∧ 𝐽 ⊆ (𝑘Gen‘𝐽)) → (𝐽t 𝐾) ⊆ ((𝑘Gen‘𝐽) ↾t 𝐾))
62, 4, 5syl2anc 587 . 2 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ⊆ ((𝑘Gen‘𝐽) ↾t 𝐾))
7 cmptop 21996 . . . . . 6 ((𝐽t 𝐾) ∈ Comp → (𝐽t 𝐾) ∈ Top)
87adantl 485 . . . . 5 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) ∈ Top)
9 restrcl 21758 . . . . . 6 ((𝐽t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V))
109simprd 499 . . . . 5 ((𝐽t 𝐾) ∈ Top → 𝐾 ∈ V)
118, 10syl 17 . . . 4 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → 𝐾 ∈ V)
12 restval 16696 . . . 4 (((𝑘Gen‘𝐽) ∈ Top ∧ 𝐾 ∈ V) → ((𝑘Gen‘𝐽) ↾t 𝐾) = ran (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)))
132, 11, 12syl2anc 587 . . 3 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → ((𝑘Gen‘𝐽) ↾t 𝐾) = ran (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)))
14 simpr 488 . . . . . 6 (((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ∈ (𝑘Gen‘𝐽))
15 simplr 768 . . . . . 6 (((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝐽t 𝐾) ∈ Comp)
16 kgeni 22138 . . . . . 6 ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽t 𝐾) ∈ Comp) → (𝑥𝐾) ∈ (𝐽t 𝐾))
1714, 15, 16syl2anc 587 . . . . 5 (((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥𝐾) ∈ (𝐽t 𝐾))
1817fmpttd 6867 . . . 4 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)):(𝑘Gen‘𝐽)⟶(𝐽t 𝐾))
1918frnd 6509 . . 3 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → ran (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥𝐾)) ⊆ (𝐽t 𝐾))
2013, 19eqsstrd 3990 . 2 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → ((𝑘Gen‘𝐽) ↾t 𝐾) ⊆ (𝐽t 𝐾))
216, 20eqssd 3969 1 ((𝐽 ∈ Top ∧ (𝐽t 𝐾) ∈ Comp) → (𝐽t 𝐾) = ((𝑘Gen‘𝐽) ↾t 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  cin 3918  wss 3919  cmpt 5132  ran crn 5543  cfv 6343  (class class class)co 7145  t crest 16690  Topctop 21494  Compccmp 21987  𝑘Genckgen 22134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-1st 7679  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-oadd 8096  df-er 8279  df-en 8500  df-fin 8503  df-fi 8866  df-rest 16692  df-topgen 16713  df-top 21495  df-topon 21512  df-bases 21547  df-cmp 21988  df-kgen 22135
This theorem is referenced by:  kgencmp2  22147  kgenidm  22148
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