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Theorem fvtresfn 6323
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 5477 . 2 (𝑋𝐵 → (𝑋𝑉) ∈ V)
2 reseq1 5422 . . 3 (𝑥 = 𝑋 → (𝑥𝑉) = (𝑋𝑉))
3 fvtresfn.f . . 3 𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))
42, 3fvmptg 6319 . 2 ((𝑋𝐵 ∧ (𝑋𝑉) ∈ V) → (𝐹𝑋) = (𝑋𝑉))
51, 4mpdan 703 1 (𝑋𝐵 → (𝐹𝑋) = (𝑋𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cmpt 4762  cres 5145  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  symgfixf1  17903  symgfixfo  17905  pwssplit1  19107  pwssplit2  19108  pwssplit3  19109  eulerpartgbij  30562  pwssplit4  37976
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