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Theorem 2ndcrest 22159
 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 22151 . . 3 (𝐽 ∈ 2ndω ↔ ∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽))
2 simplr 768 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝑥 ∈ TopBases)
3 simpll 766 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → 𝐴𝑉)
4 tgrest 21864 . . . . . . . 8 ((𝑥 ∈ TopBases ∧ 𝐴𝑉) → (topGen‘(𝑥t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴))
52, 3, 4syl2anc 587 . . . . . . 7 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥t 𝐴)) = ((topGen‘𝑥) ↾t 𝐴))
6 restbas 21863 . . . . . . . . 9 (𝑥 ∈ TopBases → (𝑥t 𝐴) ∈ TopBases)
76ad2antlr 726 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) ∈ TopBases)
8 restval 16763 . . . . . . . . . 10 ((𝑥 ∈ TopBases ∧ 𝐴𝑉) → (𝑥t 𝐴) = ran (𝑦𝑥 ↦ (𝑦𝐴)))
92, 3, 8syl2anc 587 . . . . . . . . 9 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) = ran (𝑦𝑥 ↦ (𝑦𝐴)))
10 1stcrestlem 22157 . . . . . . . . . 10 (𝑥 ≼ ω → ran (𝑦𝑥 ↦ (𝑦𝐴)) ≼ ω)
1110adantl 485 . . . . . . . . 9 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ran (𝑦𝑥 ↦ (𝑦𝐴)) ≼ ω)
129, 11eqbrtrd 5057 . . . . . . . 8 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (𝑥t 𝐴) ≼ ω)
13 2ndci 22153 . . . . . . . 8 (((𝑥t 𝐴) ∈ TopBases ∧ (𝑥t 𝐴) ≼ ω) → (topGen‘(𝑥t 𝐴)) ∈ 2ndω)
147, 12, 13syl2anc 587 . . . . . . 7 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → (topGen‘(𝑥t 𝐴)) ∈ 2ndω)
155, 14eqeltrrd 2853 . . . . . 6 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) ↾t 𝐴) ∈ 2ndω)
16 oveq1 7162 . . . . . . 7 ((topGen‘𝑥) = 𝐽 → ((topGen‘𝑥) ↾t 𝐴) = (𝐽t 𝐴))
1716eleq1d 2836 . . . . . 6 ((topGen‘𝑥) = 𝐽 → (((topGen‘𝑥) ↾t 𝐴) ∈ 2ndω ↔ (𝐽t 𝐴) ∈ 2ndω))
1815, 17syl5ibcom 248 . . . . 5 (((𝐴𝑉𝑥 ∈ TopBases) ∧ 𝑥 ≼ ω) → ((topGen‘𝑥) = 𝐽 → (𝐽t 𝐴) ∈ 2ndω))
1918expimpd 457 . . . 4 ((𝐴𝑉𝑥 ∈ TopBases) → ((𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽t 𝐴) ∈ 2ndω))
2019rexlimdva 3208 . . 3 (𝐴𝑉 → (∃𝑥 ∈ TopBases (𝑥 ≼ ω ∧ (topGen‘𝑥) = 𝐽) → (𝐽t 𝐴) ∈ 2ndω))
211, 20syl5bi 245 . 2 (𝐴𝑉 → (𝐽 ∈ 2ndω → (𝐽t 𝐴) ∈ 2ndω))
2221impcom 411 1 ((𝐽 ∈ 2ndω ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ 2ndω)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ∩ cin 3859   class class class wbr 5035   ↦ cmpt 5115  ran crn 5528  ‘cfv 6339  (class class class)co 7155  ωcom 7584   ≼ cdom 8530   ↾t crest 16757  topGenctg 16774  TopBasesctb 21650  2ndωc2ndc 22143 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-se 5487  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-er 8304  df-map 8423  df-en 8533  df-dom 8534  df-fin 8536  df-fi 8913  df-card 9406  df-acn 9409  df-rest 16759  df-topgen 16780  df-bases 21651  df-2ndc 22145 This theorem is referenced by: (None)
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