Theorem fvtresfn 6284

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 5442	. 2 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ↾ 𝑉) ∈ V)
2		reseq1 5390	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ↾ 𝑉) = (𝑋 ↾ 𝑉))
3		fvtresfn.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))
4	2, 3	fvmptg 6280	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑋 ↾ 𝑉) ∈ V) → (𝐹‘𝑋) = (𝑋 ↾ 𝑉))
5	1, 4	mpdan 702	1 ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ↦ cmpt 4729   ↾ cres 5116  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  symgfixf1  17857  symgfixfo  17859  pwssplit1  19059  pwssplit2  19060  pwssplit3  19061  eulerpartgbij  30434  pwssplit4  37659