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Theorem fvtresfn 6773
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 5901 . 2 (𝑋𝐵 → (𝑋𝑉) ∈ V)
2 reseq1 5850 . . 3 (𝑥 = 𝑋 → (𝑥𝑉) = (𝑋𝑉))
3 fvtresfn.f . . 3 𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))
42, 3fvmptg 6769 . 2 ((𝑋𝐵 ∧ (𝑋𝑉) ∈ V) → (𝐹𝑋) = (𝑋𝑉))
51, 4mpdan 685 1 (𝑋𝐵 → (𝐹𝑋) = (𝑋𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2113  Vcvv 3497  cmpt 5149  cres 5560  cfv 6358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-res 5570  df-iota 6317  df-fun 6360  df-fv 6366
This theorem is referenced by:  symgfixf1  18568  symgfixfo  18570  pwssplit1  19834  pwssplit2  19835  pwssplit3  19836  eulerpartgbij  31634  pwssplit4  39695
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