Theorem resmpt3 5946

Description: Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)

Assertion

Ref	Expression
resmpt3	⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem resmpt3

Step	Hyp	Ref	Expression
1		resres 5904	. 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ (𝐴 ∩ 𝐵))
2		ssid 3943	. . . 4 ⊢ 𝐴 ⊆ 𝐴
3		resmpt 5945	. . . 4 ⊢ (𝐴 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶))
4	2, 3	ax-mp 5	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) = (𝑥 ∈ 𝐴 ↦ 𝐶)
5	4	reseq1i 5887	. 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵)
6		inss1 4162	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
7		resmpt 5945	. . 3 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ (𝐴 ∩ 𝐵)) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶))
8	6, 7	ax-mp 5	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ (𝐴 ∩ 𝐵)) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶)
9	1, 5, 8	3eqtr3i 2774	1 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶)

Syntax hints:   = wceq 1539   ∩ cin 3886   ⊆ wss 3887   ↦ cmpt 5157   ↾ cres 5591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-mpt 5158  df-xp 5595  df-rel 5596  df-res 5601

This theorem is referenced by:  mptima  5981  offres  7826  lo1resb  15273  o1resb  15275  measinb2  32191  eulerpartgbij  32339  imassmpt  42810  limsupresicompt  43297  liminfresicompt  43321