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Theorem resmpo 7265
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 7264 . 2 ((𝐶𝐴𝐷𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐸)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝑧 = 𝐸)})
2 df-mpo 7156 . . 3 (𝑥𝐴, 𝑦𝐵𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐸)}
32reseq1i 5847 . 2 ((𝑥𝐴, 𝑦𝐵𝐸) ↾ (𝐶 × 𝐷)) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐸)} ↾ (𝐶 × 𝐷))
4 df-mpo 7156 . 2 (𝑥𝐶, 𝑦𝐷𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝑧 = 𝐸)}
51, 3, 43eqtr4g 2885 1 ((𝐶𝐴𝐷𝐵) → ((𝑥𝐴, 𝑦𝐵𝐸) ↾ (𝐶 × 𝐷)) = (𝑥𝐶, 𝑦𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2106  wss 3939   × cxp 5551  cres 5555  {coprab 7152  cmpo 7153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-opab 5125  df-xp 5559  df-rel 5560  df-res 5565  df-oprab 7155  df-mpo 7156
This theorem is referenced by:  ofmres  7679  cantnfval2  9124  pgrpsubgsymg  18458  sylow3lem5  18678  phssip  20718  mamures  20917  mdetrsca2  21129  mdetrlin2  21132  mdetunilem5  21141  smadiadetglem1  21196  smadiadetglem2  21197  pmatcollpw3lem  21307  txss12  22129  txbasval  22130  cnmpt2res  22201  fmucndlem  22815  cnmpopc  23447  oprpiece1res1  23470  oprpiece1res2  23471  cxpcn3  25242  ressplusf  30552  submatres  30958  cvmlift2lem6  32440  cvmlift2lem12  32446  icorempo  34502  elicores  41671  volicorescl  42698  rngchomrnghmresALTV  44096  rhmsubclem1  44186  rhmsubcALTVlem1  44204
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