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Theorem resmpt3 5370
 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 ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem resmpt3
StepHypRef Expression
1 resres 5329 . 2 (((𝑥𝐴𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥𝐴𝐶) ↾ (𝐴𝐵))
2 ssid 3587 . . . 4 𝐴𝐴
3 resmpt 5369 . . . 4 (𝐴𝐴 → ((𝑥𝐴𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
42, 3ax-mp 5 . . 3 ((𝑥𝐴𝐶) ↾ 𝐴) = (𝑥𝐴𝐶)
54reseq1i 5313 . 2 (((𝑥𝐴𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥𝐴𝐶) ↾ 𝐵)
6 inss1 3795 . . 3 (𝐴𝐵) ⊆ 𝐴
7 resmpt 5369 . . 3 ((𝐴𝐵) ⊆ 𝐴 → ((𝑥𝐴𝐶) ↾ (𝐴𝐵)) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶))
86, 7ax-mp 5 . 2 ((𝑥𝐴𝐶) ↾ (𝐴𝐵)) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
91, 5, 83eqtr3i 2640 1 ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475   ∩ cin 3539   ⊆ wss 3540   ↦ cmpt 4643   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-res 5050 This theorem is referenced by:  offres  7054  lo1resb  14143  o1resb  14145  measinb2  29613  eulerpartgbij  29761
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