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Theorem 2ndcrest 21477
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 21469 . . 3 (𝐽 ∈ 2nd𝜔 ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
2 simplr 744 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝑥 ∈ TopBases)
3 simpll 742 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝐴𝑉)
4 tgrest 21183 . . . . . . . 8 ((𝑥 ∈ TopBases ∧ 𝐴𝑉) → (topGen‘(𝑥t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴))
52, 3, 4syl2anc 565 . . . . . . 7 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴))
6 restbas 21182 . . . . . . . . 9 (𝑥 ∈ TopBases → (𝑥t 𝐴) ∈ TopBases)
76ad2antlr 698 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) ∈ TopBases)
8 restval 16294 . . . . . . . . . 10 ((𝑥 ∈ TopBases ∧ 𝐴𝑉) → (𝑥t 𝐴) = ran (𝑦𝑥 ↦ (𝑦𝐴)))
92, 3, 8syl2anc 565 . . . . . . . . 9 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) = ran (𝑦𝑥 ↦ (𝑦𝐴)))
10 1stcrestlem 21475 . . . . . . . . . 10 (𝑥 ≼ ω → ran (𝑦𝑥 ↦ (𝑦𝐴)) ≼ ω)
1110adantl 467 . . . . . . . . 9 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ran (𝑦𝑥 ↦ (𝑦𝐴)) ≼ ω)
129, 11eqbrtrd 4806 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) ≼ ω)
13 2ndci 21471 . . . . . . . 8 (((𝑥t 𝐴) ∈ TopBases ∧ (𝑥t 𝐴) ≼ ω) → (topGen‘(𝑥t 𝐴)) ∈ 2nd𝜔)
147, 12, 13syl2anc 565 . . . . . . 7 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥t 𝐴)) ∈ 2nd𝜔)
155, 14eqeltrrd 2850 . . . . . 6 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) ↾t 𝐴) ∈ 2nd𝜔)
16 oveq1 6799 . . . . . . 7 ((topGen‘𝑥) = 𝐽 → ((topGen‘𝑥) ↾t 𝐴) = (𝐽t 𝐴))
1716eleq1d 2834 . . . . . 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 441 . . . 4 ((𝐴𝑉𝑥 ∈ TopBases) → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽t 𝐴) ∈ 2nd𝜔))
2019rexlimdva 3178 . . 3 (𝐴𝑉 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽t 𝐴) ∈ 2nd𝜔))
211, 20syl5bi 232 . 2 (𝐴𝑉 → (𝐽 ∈ 2nd𝜔 → (𝐽t 𝐴) ∈ 2nd𝜔))
2221impcom 394 1 ((𝐽 ∈ 2nd𝜔 ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 2nd𝜔)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wrex 3061  cin 3720   class class class wbr 4784  cmpt 4861  ran crn 5250  cfv 6031  (class class class)co 6792  ωcom 7211  cdom 8106  t crest 16288  topGenctg 16305  TopBasesctb 20969  2nd𝜔c2ndc 21461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-fin 8112  df-fi 8472  df-card 8964  df-acn 8967  df-rest 16290  df-topgen 16311  df-bases 20970  df-2ndc 21463
This theorem is referenced by: (None)
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