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Theorem 2ndcrest 21197
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 21189 . . 3 (𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
2 simplr 791 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝑥 ∈ TopBases)
3 simpll 789 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝐴𝑉)
4 tgrest 20903 . . . . . . . 8 ((𝑥 ∈ TopBases ∧ 𝐴𝑉) → (topGen‘(𝑥t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴))
52, 3, 4syl2anc 692 . . . . . . 7 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴))
6 restbas 20902 . . . . . . . . 9 (𝑥 ∈ TopBases → (𝑥t 𝐴) ∈ TopBases)
76ad2antlr 762 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) ∈ TopBases)
8 restval 16027 . . . . . . . . . 10 ((𝑥 ∈ TopBases ∧ 𝐴𝑉) → (𝑥t 𝐴) = ran (𝑦𝑥 ↦ (𝑦𝐴)))
92, 3, 8syl2anc 692 . . . . . . . . 9 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) = ran (𝑦𝑥 ↦ (𝑦𝐴)))
10 1stcrestlem 21195 . . . . . . . . . 10 (𝑥 ≼ ω → ran (𝑦𝑥 ↦ (𝑦𝐴)) ≼ ω)
1110adantl 482 . . . . . . . . 9 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ran (𝑦𝑥 ↦ (𝑦𝐴)) ≼ ω)
129, 11eqbrtrd 4645 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) ≼ ω)
13 2ndci 21191 . . . . . . . 8 (((𝑥t 𝐴) ∈ TopBases ∧ (𝑥t 𝐴) ≼ ω) → (topGen‘(𝑥t 𝐴)) ∈ 2nd𝜔)
147, 12, 13syl2anc 692 . . . . . . 7 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥t 𝐴)) ∈ 2nd𝜔)
155, 14eqeltrrd 2699 . . . . . 6 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) ↾t 𝐴) ∈ 2nd𝜔)
16 oveq1 6622 . . . . . . 7 ((topGen‘𝑥) = 𝐽 → ((topGen‘𝑥) ↾t 𝐴) = (𝐽t 𝐴))
1716eleq1d 2683 . . . . . 6 ((topGen‘𝑥) = 𝐽 → (((topGen‘𝑥) ↾t 𝐴) ∈ 2nd𝜔 ↔ (𝐽t 𝐴) ∈ 2nd𝜔))
1815, 17syl5ibcom 235 . . . . 5 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) = 𝐽 → (𝐽t 𝐴) ∈ 2nd𝜔))
1918expimpd 628 . . . 4 ((𝐴𝑉𝑥 ∈ TopBases) → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽t 𝐴) ∈ 2nd𝜔))
2019rexlimdva 3026 . . 3 (𝐴𝑉 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽t 𝐴) ∈ 2nd𝜔))
211, 20syl5bi 232 . 2 (𝐴𝑉 → (𝐽 ∈ 2nd𝜔 → (𝐽t 𝐴) ∈ 2nd𝜔))
2221impcom 446 1 ((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 2nd𝜔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2909  cin 3559   class class class wbr 4623  cmpt 4683  ran crn 5085  cfv 5857  (class class class)co 6615  ωcom 7027  cdom 7913  t crest 16021  topGenctg 16038  TopBasesctb 20689  2nd𝜔c2ndc 21181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-fin 7919  df-fi 8277  df-card 8725  df-acn 8728  df-rest 16023  df-topgen 16044  df-bases 20690  df-2ndc 21183
This theorem is referenced by: (None)
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