Theorem resmpt3 5053

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 5016	. 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ (𝐴 ∩ 𝐵))
2		ssid 3244	. . . 4 ⊢ 𝐴 ⊆ 𝐴
3		resmpt 5052	. . . 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 5000	. 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵)
6		inss1 3424	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
7		resmpt 5052	. . 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 2258	1 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶)

Syntax hints:   = wceq 1395   ∩ cin 3196   ⊆ wss 3197   ↦ cmpt 4144   ↾ cres 4720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145  df-mpt 4146  df-xp 4724  df-rel 4725  df-res 4730

This theorem is referenced by:  mptima  5079  offres  6278