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Theorem restcldi 21200
Description: A closed set is closed in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
restcldi.1 𝑋 = 𝐽
Assertion
Ref Expression
restcldi ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘(𝐽t 𝐴)))

Proof of Theorem restcldi
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 simp2 1132 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘𝐽))
2 dfss 3731 . . . . 5 (𝐵𝐴𝐵 = (𝐵𝐴))
32biimpi 206 . . . 4 (𝐵𝐴𝐵 = (𝐵𝐴))
433ad2ant3 1130 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 = (𝐵𝐴))
5 ineq1 3951 . . . . 5 (𝑣 = 𝐵 → (𝑣𝐴) = (𝐵𝐴))
65eqeq2d 2771 . . . 4 (𝑣 = 𝐵 → (𝐵 = (𝑣𝐴) ↔ 𝐵 = (𝐵𝐴)))
76rspcev 3450 . . 3 ((𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵 = (𝐵𝐴)) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴))
81, 4, 7syl2anc 696 . 2 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴))
9 cldrcl 21053 . . . 4 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
1093ad2ant2 1129 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐽 ∈ Top)
11 simp1 1131 . . 3 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐴𝑋)
12 restcldi.1 . . . 4 𝑋 = 𝐽
1312restcld 21199 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑋) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
1410, 11, 13syl2anc 696 . 2 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → (𝐵 ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑣 ∈ (Clsd‘𝐽)𝐵 = (𝑣𝐴)))
158, 14mpbird 247 1 ((𝐴𝑋𝐵 ∈ (Clsd‘𝐽) ∧ 𝐵𝐴) → 𝐵 ∈ (Clsd‘(𝐽t 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1632  wcel 2140  wrex 3052  cin 3715  wss 3716   cuni 4589  cfv 6050  (class class class)co 6815  t crest 16304  Topctop 20921  Clsdccld 21043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-oadd 7735  df-er 7914  df-en 8125  df-fin 8128  df-fi 8485  df-rest 16306  df-topgen 16327  df-top 20922  df-topon 20939  df-bases 20973  df-cld 21046
This theorem is referenced by:  txkgen  21678  qtoprest  21743  cnmpt2pc  22949  cnheiborlem  22975  abelth  24415  cvmliftlem10  31605
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