fvtresfn - Metamath Proof Explorer

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Theorem fvtresfn 7010

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 6034	. 2 ⊢ (𝑋 ∈ 𝐵 → (𝑋 ↾ 𝑉) ∈ V)
2		reseq1 5981	. . 3 ⊢ (𝑥 = 𝑋 → (𝑥 ↾ 𝑉) = (𝑋 ↾ 𝑉))
3		fvtresfn.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))
4	2, 3	fvmptg 7006	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ (𝑋 ↾ 𝑉) ∈ V) → (𝐹‘𝑋) = (𝑋 ↾ 𝑉))
5	1, 4	mpdan 685	1 ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3471 ↦ cmpt 5233 ↾ cres 5682 ‘cfv 6551

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 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431

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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-res 5692 df-iota 6503 df-fun 6553 df-fv 6559

This theorem is referenced by: symgfixf1 19397 symgfixfo 19399 pwssplit1 20949 pwssplit2 20950 pwssplit3 20951 eulerpartgbij 33997 pwssplit4 42516

	
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