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Theorem ofmres 7967
Description: Equivalent expressions for a restriction of the function operation map. Unlike f 𝑅 which is a proper class, ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) can be a set by ofmresex 7968, allowing it to be used as a function or structure argument. By ofmresval 7682, the restricted operation map values are the same as the original values, allowing theorems for f 𝑅 to be reused. (Contributed by NM, 20-Oct-2014.)
Assertion
Ref Expression
ofmres ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓𝐴, 𝑔𝐵 ↦ (𝑓f 𝑅𝑔))
Distinct variable groups:   𝑓,𝑔,𝐴   𝐵,𝑓,𝑔   𝑅,𝑓,𝑔

Proof of Theorem ofmres
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssv 4001 . . 3 𝐴 ⊆ V
2 ssv 4001 . . 3 𝐵 ⊆ V
3 resmpo 7523 . . 3 ((𝐴 ⊆ V ∧ 𝐵 ⊆ V) → ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))) ↾ (𝐴 × 𝐵)) = (𝑓𝐴, 𝑔𝐵 ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))))
41, 2, 3mp2an 689 . 2 ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))) ↾ (𝐴 × 𝐵)) = (𝑓𝐴, 𝑔𝐵 ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
5 df-of 7666 . . 3 f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
65reseq1i 5970 . 2 ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) = ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))) ↾ (𝐴 × 𝐵))
7 eqid 2726 . . 3 𝐴 = 𝐴
8 eqid 2726 . . 3 𝐵 = 𝐵
9 vex 3472 . . . 4 𝑓 ∈ V
10 vex 3472 . . . 4 𝑔 ∈ V
119dmex 7898 . . . . . 6 dom 𝑓 ∈ V
1211inex1 5310 . . . . 5 (dom 𝑓 ∩ dom 𝑔) ∈ V
1312mptex 7219 . . . 4 (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) ∈ V
145ovmpt4g 7550 . . . 4 ((𝑓 ∈ V ∧ 𝑔 ∈ V ∧ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) ∈ V) → (𝑓f 𝑅𝑔) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
159, 10, 13, 14mp3an 1457 . . 3 (𝑓f 𝑅𝑔) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))
167, 8, 15mpoeq123i 7480 . 2 (𝑓𝐴, 𝑔𝐵 ↦ (𝑓f 𝑅𝑔)) = (𝑓𝐴, 𝑔𝐵 ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
174, 6, 163eqtr4i 2764 1 ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓𝐴, 𝑔𝐵 ↦ (𝑓f 𝑅𝑔))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  Vcvv 3468  cin 3942  wss 3943  cmpt 5224   × cxp 5667  dom cdm 5669  cres 5671  cfv 6536  (class class class)co 7404  cmpo 7406  f cof 7664
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7666
This theorem is referenced by:  mplsubrglem  21901  psrplusgpropd  22105  ofoafg  42661  naddcnff  42669
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