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Theorem fvtresfn 7031
Description: Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypothesis
Ref Expression
fvtresfn.f 𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))
Assertion
Ref Expression
fvtresfn (𝑋𝐵 → (𝐹𝑋) = (𝑋𝑉))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑉   𝑥,𝑋
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem fvtresfn
StepHypRef Expression
1 resexg 6056 . 2 (𝑋𝐵 → (𝑋𝑉) ∈ V)
2 reseq1 6003 . . 3 (𝑥 = 𝑋 → (𝑥𝑉) = (𝑋𝑉))
3 fvtresfn.f . . 3 𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))
42, 3fvmptg 7027 . 2 ((𝑋𝐵 ∧ (𝑋𝑉) ∈ V) → (𝐹𝑋) = (𝑋𝑉))
51, 4mpdan 686 1 (𝑋𝐵 → (𝐹𝑋) = (𝑋𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  Vcvv 3488  cmpt 5249  cres 5702  cfv 6573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581
This theorem is referenced by:  symgfixf1  19479  symgfixfo  19481  pwssplit1  21081  pwssplit2  21082  pwssplit3  21083  eulerpartgbij  34337  pwssplit4  43046
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