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Theorem restval 17471
Description: The subspace topology induced by the topology 𝐽 on the set 𝐴. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
restval ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem restval
Dummy variables 𝑗 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3501 . 2 (𝐽𝑉𝐽 ∈ V)
2 elex 3501 . 2 (𝐴𝑊𝐴 ∈ V)
3 mptexg 7241 . . . . 5 (𝐽 ∈ V → (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
4 rnexg 7924 . . . . 5 ((𝑥𝐽 ↦ (𝑥𝐴)) ∈ V → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
53, 4syl 17 . . . 4 (𝐽 ∈ V → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
65adantr 480 . . 3 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
7 simpl 482 . . . . . 6 ((𝑗 = 𝐽𝑦 = 𝐴) → 𝑗 = 𝐽)
8 simpr 484 . . . . . . 7 ((𝑗 = 𝐽𝑦 = 𝐴) → 𝑦 = 𝐴)
98ineq2d 4220 . . . . . 6 ((𝑗 = 𝐽𝑦 = 𝐴) → (𝑥𝑦) = (𝑥𝐴))
107, 9mpteq12dv 5233 . . . . 5 ((𝑗 = 𝐽𝑦 = 𝐴) → (𝑥𝑗 ↦ (𝑥𝑦)) = (𝑥𝐽 ↦ (𝑥𝐴)))
1110rneqd 5949 . . . 4 ((𝑗 = 𝐽𝑦 = 𝐴) → ran (𝑥𝑗 ↦ (𝑥𝑦)) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
12 df-rest 17467 . . . 4 t = (𝑗 ∈ V, 𝑦 ∈ V ↦ ran (𝑥𝑗 ↦ (𝑥𝑦)))
1311, 12ovmpoga 7587 . . 3 ((𝐽 ∈ V ∧ 𝐴 ∈ V ∧ ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
146, 13mpd3an3 1464 . 2 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
151, 2, 14syl2an 596 1 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  cmpt 5225  ran crn 5686  (class class class)co 7431  t crest 17465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17467
This theorem is referenced by:  elrest  17472  0rest  17474  restid2  17475  tgrest  23167  resttopon  23169  restco  23172  rest0  23177  restfpw  23187  neitr  23188  ordtrest2  23212  1stcrest  23461  2ndcrest  23462  kgencmp  23553  xkoptsub  23662  trfilss  23897  trfg  23899  uzrest  23905  restmetu  24583  ellimc2  25912  limcflf  25916  ordtrest2NEW  33922  ptrest  37626
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