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Theorem restcls 21185
Description: A closure in a subspace topology. (Contributed by Jeff Hankins, 22-Jan-2010.) (Revised by Mario Carneiro, 15-Dec-2013.)
Hypotheses
Ref Expression
restcls.1 𝑋 = 𝐽
restcls.2 𝐾 = (𝐽t 𝑌)
Assertion
Ref Expression
restcls ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))

Proof of Theorem restcls
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simp1 1131 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐽 ∈ Top)
2 sstr 3750 . . . . . . . 8 ((𝑆𝑌𝑌𝑋) → 𝑆𝑋)
32ancoms 468 . . . . . . 7 ((𝑌𝑋𝑆𝑌) → 𝑆𝑋)
433adant1 1125 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑋)
5 restcls.1 . . . . . . 7 𝑋 = 𝐽
65clscld 21051 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
71, 4, 6syl2anc 696 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽))
8 eqid 2758 . . . . 5 (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)
9 ineq1 3948 . . . . . . 7 (𝑥 = ((cls‘𝐽)‘𝑆) → (𝑥𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
109eqeq2d 2768 . . . . . 6 (𝑥 = ((cls‘𝐽)‘𝑆) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)))
1110rspcev 3447 . . . . 5 ((((cls‘𝐽)‘𝑆) ∈ (Clsd‘𝐽) ∧ (((cls‘𝐽)‘𝑆) ∩ 𝑌) = (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌))
127, 8, 11sylancl 697 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌))
13 restcls.2 . . . . . . 7 𝐾 = (𝐽t 𝑌)
1413fveq2i 6353 . . . . . 6 (Clsd‘𝐾) = (Clsd‘(𝐽t 𝑌))
1514eleq2i 2829 . . . . 5 ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)))
165restcld 21176 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
17163adant3 1127 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
1815, 17syl5bb 272 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(((cls‘𝐽)‘𝑆) ∩ 𝑌) = (𝑥𝑌)))
1912, 18mpbird 247 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾))
205sscls 21060 . . . . 5 ((𝐽 ∈ Top ∧ 𝑆𝑋) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
211, 4, 20syl2anc 696 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐽)‘𝑆))
22 simp3 1133 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆𝑌)
2321, 22ssind 3978 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
24 eqid 2758 . . . 4 𝐾 = 𝐾
2524clsss2 21076 . . 3 (((((cls‘𝐽)‘𝑆) ∩ 𝑌) ∈ (Clsd‘𝐾) ∧ 𝑆 ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌)) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
2619, 23, 25syl2anc 696 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) ⊆ (((cls‘𝐽)‘𝑆) ∩ 𝑌))
2713fveq2i 6353 . . . . . 6 (cls‘𝐾) = (cls‘(𝐽t 𝑌))
2827fveq1i 6351 . . . . 5 ((cls‘𝐾)‘𝑆) = ((cls‘(𝐽t 𝑌))‘𝑆)
29 id 22 . . . . . . . . 9 (𝑌𝑋𝑌𝑋)
305topopn 20911 . . . . . . . . 9 (𝐽 ∈ Top → 𝑋𝐽)
31 ssexg 4954 . . . . . . . . 9 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
3229, 30, 31syl2anr 496 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 ∈ V)
33 resttop 21164 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝐽t 𝑌) ∈ Top)
3432, 33syldan 488 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (𝐽t 𝑌) ∈ Top)
35343adant3 1127 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (𝐽t 𝑌) ∈ Top)
365restuni 21166 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝑌 = (𝐽t 𝑌))
37363adant3 1127 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑌 = (𝐽t 𝑌))
3822, 37sseqtrd 3780 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 (𝐽t 𝑌))
39 eqid 2758 . . . . . . 7 (𝐽t 𝑌) = (𝐽t 𝑌)
4039clscld 21051 . . . . . 6 (((𝐽t 𝑌) ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → ((cls‘(𝐽t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
4135, 38, 40syl2anc 696 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘(𝐽t 𝑌))‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
4228, 41syl5eqel 2841 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)))
435restcld 21176 . . . . 5 ((𝐽 ∈ Top ∧ 𝑌𝑋) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌)))
44433adant3 1127 . . . 4 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐾)‘𝑆) ∈ (Clsd‘(𝐽t 𝑌)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌)))
4542, 44mpbid 222 . . 3 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ∃𝑥 ∈ (Clsd‘𝐽)((cls‘𝐾)‘𝑆) = (𝑥𝑌))
4613, 34syl5eqel 2841 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑌𝑋) → 𝐾 ∈ Top)
47463adant3 1127 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝐾 ∈ Top)
4813unieqi 4595 . . . . . . . . 9 𝐾 = (𝐽t 𝑌)
4948eqcomi 2767 . . . . . . . 8 (𝐽t 𝑌) = 𝐾
5049sscls 21060 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑆 (𝐽t 𝑌)) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
5147, 38, 50syl2anc 696 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
5251adantr 472 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → 𝑆 ⊆ ((cls‘𝐾)‘𝑆))
53 inss1 3974 . . . . . . 7 (𝑥𝑌) ⊆ 𝑥
54 sseq1 3765 . . . . . . 7 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐾)‘𝑆) ⊆ 𝑥 ↔ (𝑥𝑌) ⊆ 𝑥))
5553, 54mpbiri 248 . . . . . 6 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
5655ad2antll 767 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → ((cls‘𝐾)‘𝑆) ⊆ 𝑥)
5752, 56sstrd 3752 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → 𝑆𝑥)
585clsss2 21076 . . . . . . . . . 10 ((𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
5958adantl 473 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → ((cls‘𝐽)‘𝑆) ⊆ 𝑥)
60 ssrin 3979 . . . . . . . . 9 (((cls‘𝐽)‘𝑆) ⊆ 𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥𝑌))
6159, 60syl 17 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥𝑌))
62 sseq2 3766 . . . . . . . 8 (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → ((((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆) ↔ (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ (𝑥𝑌)))
6361, 62syl5ibrcom 237 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ 𝑆𝑥)) → (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))
6463expr 644 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑆𝑥 → (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))))
6564com23 86 . . . . 5 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ 𝑥 ∈ (Clsd‘𝐽)) → (((cls‘𝐾)‘𝑆) = (𝑥𝑌) → (𝑆𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))))
6665impr 650 . . . 4 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → (𝑆𝑥 → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆)))
6757, 66mpd 15 . . 3 (((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) ∧ (𝑥 ∈ (Clsd‘𝐽) ∧ ((cls‘𝐾)‘𝑆) = (𝑥𝑌))) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
6845, 67rexlimddv 3171 . 2 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → (((cls‘𝐽)‘𝑆) ∩ 𝑌) ⊆ ((cls‘𝐾)‘𝑆))
6926, 68eqssd 3759 1 ((𝐽 ∈ Top ∧ 𝑌𝑋𝑆𝑌) → ((cls‘𝐾)‘𝑆) = (((cls‘𝐽)‘𝑆) ∩ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1630  wcel 2137  wrex 3049  Vcvv 3338  cin 3712  wss 3713   cuni 4586  cfv 6047  (class class class)co 6811  t crest 16281  Topctop 20898  Clsdccld 21020  clsccl 21022
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 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-oadd 7731  df-er 7909  df-en 8120  df-fin 8123  df-fi 8480  df-rest 16283  df-topgen 16304  df-top 20899  df-topon 20916  df-bases 20950  df-cld 21023  df-cls 21025
This theorem is referenced by:  restlp  21187  resscdrg  23352
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