Theorem resmpt3 4868

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 4831	. 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ (𝐴 ∩ 𝐵))
2		ssid 3117	. . . 4 ⊢ 𝐴 ⊆ 𝐴
3		resmpt 4867	. . . 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 4815	. 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵)
6		inss1 3296	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
7		resmpt 4867	. . 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 2168	1 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶)

Syntax hints:   = wceq 1331   ∩ cin 3070   ⊆ wss 3071   ↦ cmpt 3989   ↾ cres 4541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-mpt 3991  df-xp 4545  df-rel 4546  df-res 4551

This theorem is referenced by:  offres  6033