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Theorem 2ndcrest 23280
Description: A subspace of a second-countable space is second-countable. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
2ndcrest ((𝐽 ∈ 2ndω ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 2ndω)

Proof of Theorem 2ndcrest
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 is2ndc 23272 . . 3 (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
2 simplr 766 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝑥 ∈ TopBases)
3 simpll 764 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝐴𝑉)
4 tgrest 22985 . . . . . . . 8 ((𝑥 ∈ TopBases ∧ 𝐴𝑉) → (topGen‘(𝑥t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴))
52, 3, 4syl2anc 583 . . . . . . 7 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴))
6 restbas 22984 . . . . . . . . 9 (𝑥 ∈ TopBases → (𝑥t 𝐴) ∈ TopBases)
76ad2antlr 724 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) ∈ TopBases)
8 restval 17371 . . . . . . . . . 10 ((𝑥 ∈ TopBases ∧ 𝐴𝑉) → (𝑥t 𝐴) = ran (𝑦𝑥 ↦ (𝑦𝐴)))
92, 3, 8syl2anc 583 . . . . . . . . 9 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) = ran (𝑦𝑥 ↦ (𝑦𝐴)))
10 1stcrestlem 23278 . . . . . . . . . 10 (𝑥 ≼ ω → ran (𝑦𝑥 ↦ (𝑦𝐴)) ≼ ω)
1110adantl 481 . . . . . . . . 9 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ran (𝑦𝑥 ↦ (𝑦𝐴)) ≼ ω)
129, 11eqbrtrd 5160 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) ≼ ω)
13 2ndci 23274 . . . . . . . 8 (((𝑥t 𝐴) ∈ TopBases ∧ (𝑥t 𝐴) ≼ ω) → (topGen‘(𝑥t 𝐴)) ∈ 2ndω)
147, 12, 13syl2anc 583 . . . . . . 7 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥t 𝐴)) ∈ 2ndω)
155, 14eqeltrrd 2826 . . . . . 6 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) ↾t 𝐴) ∈ 2ndω)
16 oveq1 7408 . . . . . . 7 ((topGen‘𝑥) = 𝐽 → ((topGen‘𝑥) ↾t 𝐴) = (𝐽t 𝐴))
1716eleq1d 2810 . . . . . 6 ((topGen‘𝑥) = 𝐽 → (((topGen‘𝑥) ↾t 𝐴) ∈ 2ndω ↔ (𝐽t 𝐴) ∈ 2ndω))
1815, 17syl5ibcom 244 . . . . 5 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) = 𝐽 → (𝐽t 𝐴) ∈ 2ndω))
1918expimpd 453 . . . 4 ((𝐴𝑉𝑥 ∈ TopBases) → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽t 𝐴) ∈ 2ndω))
2019rexlimdva 3147 . . 3 (𝐴𝑉 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽t 𝐴) ∈ 2ndω))
211, 20biimtrid 241 . 2 (𝐴𝑉 → (𝐽 ∈ 2ndω → (𝐽t 𝐴) ∈ 2ndω))
2221impcom 407 1 ((𝐽 ∈ 2ndω ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 2ndω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wrex 3062  cin 3939   class class class wbr 5138  cmpt 5221  ran crn 5667  cfv 6533  (class class class)co 7401  ωcom 7848  cdom 8933  t crest 17365  topGenctg 17382  TopBasesctb 22770  2ndωc2ndc 23264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-fin 8939  df-fi 9402  df-card 9930  df-acn 9933  df-rest 17367  df-topgen 17388  df-bases 22771  df-2ndc 23266
This theorem is referenced by: (None)
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