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Theorem resmpt3 5608
 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 5567 . 2 (((𝑥𝐴𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥𝐴𝐶) ↾ (𝐴𝐵))
2 ssid 3765 . . . 4 𝐴𝐴
3 resmpt 5607 . . . 4 (𝐴𝐴 → ((𝑥𝐴𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
42, 3ax-mp 5 . . 3 ((𝑥𝐴𝐶) ↾ 𝐴) = (𝑥𝐴𝐶)
54reseq1i 5547 . 2 (((𝑥𝐴𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥𝐴𝐶) ↾ 𝐵)
6 inss1 3976 . . 3 (𝐴𝐵) ⊆ 𝐴
7 resmpt 5607 . . 3 ((𝐴𝐵) ⊆ 𝐴 → ((𝑥𝐴𝐶) ↾ (𝐴𝐵)) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶))
86, 7ax-mp 5 . 2 ((𝑥𝐴𝐶) ↾ (𝐴𝐵)) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
91, 5, 83eqtr3i 2790 1 ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∩ cin 3714   ⊆ wss 3715   ↦ cmpt 4881   ↾ cres 5268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-mpt 4882  df-xp 5272  df-rel 5273  df-res 5278 This theorem is referenced by:  offres  7329  lo1resb  14514  o1resb  14516  measinb2  30616  eulerpartgbij  30764  mptima  39954  imassmpt  39998  limsupresicompt  40509  liminfresicompt  40533
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