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Theorem fvtresfn 6178
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 5349 . 2 (𝑋𝐵 → (𝑋𝑉) ∈ V)
2 reseq1 5298 . . 3 (𝑥 = 𝑋 → (𝑥𝑉) = (𝑋𝑉))
3 fvtresfn.f . . 3 𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))
42, 3fvmptg 6174 . 2 ((𝑋𝐵 ∧ (𝑋𝑉) ∈ V) → (𝐹𝑋) = (𝑋𝑉))
51, 4mpdan 698 1 (𝑋𝐵 → (𝐹𝑋) = (𝑋𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3172  cmpt 4637  cres 5030  cfv 5790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-res 5040  df-iota 5754  df-fun 5792  df-fv 5798
This theorem is referenced by:  symgfixf1  17626  symgfixfo  17628  pwssplit1  18826  pwssplit2  18827  pwssplit3  18828  eulerpartgbij  29567  pwssplit4  36473
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