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Theorem restval 11826
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 2644 . 2 (𝐽𝑉𝐽 ∈ V)
2 elex 2644 . 2 (𝐴𝑊𝐴 ∈ V)
3 mptexg 5561 . . . . 5 (𝐽 ∈ V → (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
4 rnexg 4730 . . . . 5 ((𝑥𝐽 ↦ (𝑥𝐴)) ∈ V → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
53, 4syl 14 . . . 4 (𝐽 ∈ V → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
65adantr 271 . . 3 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
7 simpl 108 . . . . . 6 ((𝑗 = 𝐽𝑦 = 𝐴) → 𝑗 = 𝐽)
8 simpr 109 . . . . . . 7 ((𝑗 = 𝐽𝑦 = 𝐴) → 𝑦 = 𝐴)
98ineq2d 3216 . . . . . 6 ((𝑗 = 𝐽𝑦 = 𝐴) → (𝑥𝑦) = (𝑥𝐴))
107, 9mpteq12dv 3942 . . . . 5 ((𝑗 = 𝐽𝑦 = 𝐴) → (𝑥𝑗 ↦ (𝑥𝑦)) = (𝑥𝐽 ↦ (𝑥𝐴)))
1110rneqd 4696 . . . 4 ((𝑗 = 𝐽𝑦 = 𝐴) → ran (𝑥𝑗 ↦ (𝑥𝑦)) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
12 df-rest 11822 . . . 4 t = (𝑗 ∈ V, 𝑦 ∈ V ↦ ran (𝑥𝑗 ↦ (𝑥𝑦)))
1311, 12ovmpt2ga 5812 . . 3 ((𝐽 ∈ V ∧ 𝐴 ∈ V ∧ ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
146, 13mpd3an3 1281 . 2 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
151, 2, 14syl2an 284 1 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wcel 1445  Vcvv 2633  cin 3012  cmpt 3921  ran crn 4468  (class class class)co 5690  t crest 11820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-rest 11822
This theorem is referenced by:  elrest  11827  restid2  11829  tgrest  12037  resttopon  12039  restco  12042  rest0  12047
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