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Theorem resttopon2 21776
Description: The underlying set of a subspace topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
resttopon2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ (TopOn‘(𝐴𝑋)))

Proof of Theorem resttopon2
StepHypRef Expression
1 topontop 21521 . . 3 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2 resttop 21768 . . 3 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
31, 2sylan 583 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ Top)
4 toponuni 21522 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
54ineq2d 4174 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝐴𝑋) = (𝐴 𝐽))
65adantr 484 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐴𝑋) = (𝐴 𝐽))
7 eqid 2824 . . . . 5 𝐽 = 𝐽
87restuni2 21775 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
91, 8sylan 583 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐴 𝐽) = (𝐽t 𝐴))
106, 9eqtrd 2859 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐴𝑋) = (𝐽t 𝐴))
11 istopon 21520 . 2 ((𝐽t 𝐴) ∈ (TopOn‘(𝐴𝑋)) ↔ ((𝐽t 𝐴) ∈ Top ∧ (𝐴𝑋) = (𝐽t 𝐴)))
123, 10, 11sylanbrc 586 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑉) → (𝐽t 𝐴) ∈ (TopOn‘(𝐴𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cin 3918   cuni 4824  cfv 6343  (class class class)co 7149  t crest 16694  Topctop 21501  TopOnctopon 21518
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-oadd 8102  df-er 8285  df-en 8506  df-fin 8509  df-fi 8872  df-rest 16696  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554
This theorem is referenced by:  resstps  21795  lmss  21906
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