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Theorem restval 12747
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 2763 . 2 (𝐽𝑉𝐽 ∈ V)
2 elex 2763 . 2 (𝐴𝑊𝐴 ∈ V)
3 mptexg 5761 . . . . 5 (𝐽 ∈ V → (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
4 rnexg 4910 . . . . 5 ((𝑥𝐽 ↦ (𝑥𝐴)) ∈ V → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
53, 4syl 14 . . . 4 (𝐽 ∈ V → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
65adantr 276 . . 3 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
7 simpl 109 . . . . . 6 ((𝑗 = 𝐽𝑦 = 𝐴) → 𝑗 = 𝐽)
8 simpr 110 . . . . . . 7 ((𝑗 = 𝐽𝑦 = 𝐴) → 𝑦 = 𝐴)
98ineq2d 3351 . . . . . 6 ((𝑗 = 𝐽𝑦 = 𝐴) → (𝑥𝑦) = (𝑥𝐴))
107, 9mpteq12dv 4100 . . . . 5 ((𝑗 = 𝐽𝑦 = 𝐴) → (𝑥𝑗 ↦ (𝑥𝑦)) = (𝑥𝐽 ↦ (𝑥𝐴)))
1110rneqd 4874 . . . 4 ((𝑗 = 𝐽𝑦 = 𝐴) → ran (𝑥𝑗 ↦ (𝑥𝑦)) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
12 df-rest 12743 . . . 4 t = (𝑗 ∈ V, 𝑦 ∈ V ↦ ran (𝑥𝑗 ↦ (𝑥𝑦)))
1311, 12ovmpoga 6025 . . 3 ((𝐽 ∈ V ∧ 𝐴 ∈ V ∧ ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
146, 13mpd3an3 1349 . 2 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
151, 2, 14syl2an 289 1 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  Vcvv 2752  cin 3143  cmpt 4079  ran crn 4645  (class class class)co 5895  t crest 12741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-rest 12743
This theorem is referenced by:  elrest  12748  restid2  12750  tgrest  14121  resttopon  14123  restco  14126  rest0  14131
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