Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  resmpt GIF version

Theorem resmpt 4656
 Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
Assertion
Ref Expression
resmpt (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem resmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 resopab2 4655 . 2 (𝐵𝐴 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ↾ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)})
2 df-mpt 3820 . . 3 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
32reseq1i 4608 . 2 ((𝑥𝐴𝐶) ↾ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ↾ 𝐵)
4 df-mpt 3820 . 2 (𝑥𝐵𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦 = 𝐶)}
51, 3, 43eqtr4g 2097 1 (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393   ⊆ wss 2917  {copab 3817   ↦ cmpt 3818   ↾ cres 4347 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-res 4357 This theorem is referenced by:  resmpt3  4657  f1stres  5786  f2ndres  5787  tposss  5861  dftpos2  5876  dftpos4  5878
 Copyright terms: Public domain W3C validator