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Theorem 2ndcrest 21997
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 21989 . . 3 (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
2 simplr 765 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝑥 ∈ TopBases)
3 simpll 763 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝐴𝑉)
4 tgrest 21702 . . . . . . . 8 ((𝑥 ∈ TopBases ∧ 𝐴𝑉) → (topGen‘(𝑥t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴))
52, 3, 4syl2anc 584 . . . . . . 7 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴))
6 restbas 21701 . . . . . . . . 9 (𝑥 ∈ TopBases → (𝑥t 𝐴) ∈ TopBases)
76ad2antlr 723 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) ∈ TopBases)
8 restval 16695 . . . . . . . . . 10 ((𝑥 ∈ TopBases ∧ 𝐴𝑉) → (𝑥t 𝐴) = ran (𝑦𝑥 ↦ (𝑦𝐴)))
92, 3, 8syl2anc 584 . . . . . . . . 9 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) = ran (𝑦𝑥 ↦ (𝑦𝐴)))
10 1stcrestlem 21995 . . . . . . . . . 10 (𝑥 ≼ ω → ran (𝑦𝑥 ↦ (𝑦𝐴)) ≼ ω)
1110adantl 482 . . . . . . . . 9 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ran (𝑦𝑥 ↦ (𝑦𝐴)) ≼ ω)
129, 11eqbrtrd 5085 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) ≼ ω)
13 2ndci 21991 . . . . . . . 8 (((𝑥t 𝐴) ∈ TopBases ∧ (𝑥t 𝐴) ≼ ω) → (topGen‘(𝑥t 𝐴)) ∈ 2ndω)
147, 12, 13syl2anc 584 . . . . . . 7 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥t 𝐴)) ∈ 2ndω)
155, 14eqeltrrd 2919 . . . . . 6 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) ↾t 𝐴) ∈ 2ndω)
16 oveq1 7157 . . . . . . 7 ((topGen‘𝑥) = 𝐽 → ((topGen‘𝑥) ↾t 𝐴) = (𝐽t 𝐴))
1716eleq1d 2902 . . . . . 6 ((topGen‘𝑥) = 𝐽 → (((topGen‘𝑥) ↾t 𝐴) ∈ 2ndω ↔ (𝐽t 𝐴) ∈ 2ndω))
1815, 17syl5ibcom 246 . . . . 5 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) = 𝐽 → (𝐽t 𝐴) ∈ 2ndω))
1918expimpd 454 . . . 4 ((𝐴𝑉𝑥 ∈ TopBases) → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽t 𝐴) ∈ 2ndω))
2019rexlimdva 3289 . . 3 (𝐴𝑉 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽t 𝐴) ∈ 2ndω))
211, 20syl5bi 243 . 2 (𝐴𝑉 → (𝐽 ∈ 2ndω → (𝐽t 𝐴) ∈ 2ndω))
2221impcom 408 1 ((𝐽 ∈ 2ndω ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 2ndω)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wrex 3144  cin 3939   class class class wbr 5063  cmpt 5143  ran crn 5555  cfv 6354  (class class class)co 7150  ωcom 7573  cdom 8501  t crest 16689  topGenctg 16706  TopBasesctb 21488  2ndωc2ndc 21981
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-fin 8507  df-fi 8869  df-card 9362  df-acn 9365  df-rest 16691  df-topgen 16712  df-bases 21489  df-2ndc 21983
This theorem is referenced by: (None)
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