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Theorem resmpt3 5877
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 5835 . 2 (((𝑥𝐴𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥𝐴𝐶) ↾ (𝐴𝐵))
2 ssid 3940 . . . 4 𝐴𝐴
3 resmpt 5876 . . . 4 (𝐴𝐴 → ((𝑥𝐴𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
42, 3ax-mp 5 . . 3 ((𝑥𝐴𝐶) ↾ 𝐴) = (𝑥𝐴𝐶)
54reseq1i 5818 . 2 (((𝑥𝐴𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥𝐴𝐶) ↾ 𝐵)
6 inss1 4158 . . 3 (𝐴𝐵) ⊆ 𝐴
7 resmpt 5876 . . 3 ((𝐴𝐵) ⊆ 𝐴 → ((𝑥𝐴𝐶) ↾ (𝐴𝐵)) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶))
86, 7ax-mp 5 . 2 ((𝑥𝐴𝐶) ↾ (𝐴𝐵)) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
91, 5, 83eqtr3i 2832 1 ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cin 3883  wss 3884  cmpt 5113  cres 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096  df-mpt 5114  df-xp 5529  df-rel 5530  df-res 5535
This theorem is referenced by:  mptima  5912  offres  7670  lo1resb  14917  o1resb  14919  measinb2  31596  eulerpartgbij  31744  imassmpt  41899  limsupresicompt  42395  liminfresicompt  42419
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