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Theorem resmpo 5972
Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)
Assertion
Ref Expression
resmpo ((𝐶𝐴𝐷𝐵) → ((𝑥𝐴, 𝑦𝐵𝐸) ↾ (𝐶 × 𝐷)) = (𝑥𝐶, 𝑦𝐷𝐸))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem resmpo
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 resoprab2 5971 . 2 ((𝐶𝐴𝐷𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐸)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝑧 = 𝐸)})
2 df-mpo 5879 . . 3 (𝑥𝐴, 𝑦𝐵𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐸)}
32reseq1i 4903 . 2 ((𝑥𝐴, 𝑦𝐵𝐸) ↾ (𝐶 × 𝐷)) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐸)} ↾ (𝐶 × 𝐷))
4 df-mpo 5879 . 2 (𝑥𝐶, 𝑦𝐷𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝑧 = 𝐸)}
51, 3, 43eqtr4g 2235 1 ((𝐶𝐴𝐷𝐵) → ((𝑥𝐴, 𝑦𝐵𝐸) ↾ (𝐶 × 𝐷)) = (𝑥𝐶, 𝑦𝐷𝐸))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wss 3129   × cxp 4624  cres 4628  {coprab 5875  cmpo 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4065  df-xp 4632  df-rel 4633  df-res 4638  df-oprab 5878  df-mpo 5879
This theorem is referenced by:  ofmres  6136  divfnzn  9619  txss12  13697  txbasval  13698  cnmpt2res  13728
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