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Theorem resmpt3 4685
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 4652 . 2 (((𝑥𝐴𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥𝐴𝐶) ↾ (𝐴𝐵))
2 ssid 2992 . . . 4 𝐴𝐴
3 resmpt 4684 . . . 4 (𝐴𝐴 → ((𝑥𝐴𝐶) ↾ 𝐴) = (𝑥𝐴𝐶))
42, 3ax-mp 7 . . 3 ((𝑥𝐴𝐶) ↾ 𝐴) = (𝑥𝐴𝐶)
54reseq1i 4636 . 2 (((𝑥𝐴𝐶) ↾ 𝐴) ↾ 𝐵) = ((𝑥𝐴𝐶) ↾ 𝐵)
6 inss1 3185 . . 3 (𝐴𝐵) ⊆ 𝐴
7 resmpt 4684 . . 3 ((𝐴𝐵) ⊆ 𝐴 → ((𝑥𝐴𝐶) ↾ (𝐴𝐵)) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶))
86, 7ax-mp 7 . 2 ((𝑥𝐴𝐶) ↾ (𝐴𝐵)) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
91, 5, 83eqtr3i 2084 1 ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥 ∈ (𝐴𝐵) ↦ 𝐶)
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cin 2944  wss 2945  cmpt 3846  cres 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-mpt 3848  df-xp 4379  df-rel 4380  df-res 4385
This theorem is referenced by:  offres  5790
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