Theorem fvtresfn 7006

Description: Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Hypothesis

Ref	Expression
fvtresfn.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))

Assertion

Ref	Expression
fvtresfn	⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑉   𝑥,𝑋

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem fvtresfn

Step	Hyp	Ref	Expression
1		resexg 6032	. 2 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ↾ 𝑉) ∈ V)
2		reseq1 5979	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ↾ 𝑉) = (𝑋 ↾ 𝑉))
3		fvtresfn.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))
4	2, 3	fvmptg 7002	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑋 ↾ 𝑉) ∈ V) → (𝐹‘𝑋) = (𝑋 ↾ 𝑉))
5	1, 4	mpdan 685	1 ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ↦ cmpt 5232   ↾ cres 5680  ‘cfv 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-iota 6501  df-fun 6551  df-fv 6557

This theorem is referenced by:  symgfixf1  19404  symgfixfo  19406  pwssplit1  20956  pwssplit2  20957  pwssplit3  20958  eulerpartgbij  34123  pwssplit4  42655