Theorem resmpo 7531

Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)

Assertion

Ref	Expression
resmpo	⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸))

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.

Step	Hyp	Ref	Expression
1		resoprab2 7530	. 2 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐸)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝑧 = 𝐸)})
2		df-mpo 7416	. . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐸)}
3	2	reseq1i 5975	. 2 ⊢ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐸)} ↾ (𝐶 × 𝐷))
4		df-mpo 7416	. 2 ⊢ (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝑧 = 𝐸)}
5	1, 3, 4	3eqtr4g 2829	1 ⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913   × cxp 5660   ↾ cres 5664  {coprab 7412   ∈ cmpo 7413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-opab 5178  df-xp 5668  df-rel 5669  df-res 5674  df-oprab 7415  df-mpo 7416

This theorem is referenced by:  elimampo  7548  ofmres  7980  cantnfval2  9637  submefmnd  18953  pgrpsubgsymg  19478  sylow3lem5  19700  rhmsubclem1  20769  phssip  21776  mamures  22522  mdetrsca2  22729  mdetrlin2  22732  mdetunilem5  22741  smadiadetglem1  22796  smadiadetglem2  22797  pmatcollpw3lem  22908  txss12  23730  txbasval  23731  cnmpt2res  23802  fmucndlem  24415  cnmpopc  25055  oprpiece1res1  25078  oprpiece1res2  25079  cxpcn3  26878  ressplusf  33223  submatres  34140  cvmlift2lem6  35698  cvmlift2lem12  35704  icorempo  37884  elicores  46140  volicorescl  47158  rngchomrnghmresALTV  48932  rhmsubcALTVlem1  48934  rescofuf  49755