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Theorem restval 12572
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 2741 . 2 (𝐽𝑉𝐽 ∈ V)
2 elex 2741 . 2 (𝐴𝑊𝐴 ∈ V)
3 mptexg 5718 . . . . 5 (𝐽 ∈ V → (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
4 rnexg 4874 . . . . 5 ((𝑥𝐽 ↦ (𝑥𝐴)) ∈ V → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
53, 4syl 14 . . . 4 (𝐽 ∈ V → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
65adantr 274 . . 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 3328 . . . . . 6 ((𝑗 = 𝐽𝑦 = 𝐴) → (𝑥𝑦) = (𝑥𝐴))
107, 9mpteq12dv 4069 . . . . 5 ((𝑗 = 𝐽𝑦 = 𝐴) → (𝑥𝑗 ↦ (𝑥𝑦)) = (𝑥𝐽 ↦ (𝑥𝐴)))
1110rneqd 4838 . . . 4 ((𝑗 = 𝐽𝑦 = 𝐴) → ran (𝑥𝑗 ↦ (𝑥𝑦)) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
12 df-rest 12568 . . . 4 t = (𝑗 ∈ V, 𝑦 ∈ V ↦ ran (𝑥𝑗 ↦ (𝑥𝑦)))
1311, 12ovmpoga 5979 . . 3 ((𝐽 ∈ V ∧ 𝐴 ∈ V ∧ ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
146, 13mpd3an3 1333 . 2 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
151, 2, 14syl2an 287 1 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  Vcvv 2730  cin 3120  cmpt 4048  ran crn 4610  (class class class)co 5850  t crest 12566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-rest 12568
This theorem is referenced by:  elrest  12573  restid2  12575  tgrest  12922  resttopon  12924  restco  12927  rest0  12932
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