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Theorem ofmres 5790
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 5791, allowing it to be used as a function or structure argument. By ofmresval 5750, 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 2992 . . 3 𝐴 ⊆ V
2 ssv 2992 . . 3 𝐵 ⊆ V
3 resmpt2 5626 . . 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 410 . 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 5739 . . 3 𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
65reseq1i 4635 . 2 ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = ((𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))) ↾ (𝐴 × 𝐵))
7 eqid 2056 . . 3 𝐴 = 𝐴
8 eqid 2056 . . 3 𝐵 = 𝐵
9 vex 2577 . . . 4 𝑓 ∈ V
10 vex 2577 . . . 4 𝑔 ∈ V
119dmex 4625 . . . . . 6 dom 𝑓 ∈ V
1211inex1 3918 . . . . 5 (dom 𝑓 ∩ dom 𝑔) ∈ V
1312mptex 5414 . . . 4 (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) ∈ V
145ovmpt4g 5650 . . . 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 1243 . . 3 (𝑓𝑓 𝑅𝑔) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))
167, 8, 15mpt2eq123i 5595 . 2 (𝑓𝐴, 𝑔𝐵 ↦ (𝑓𝑓 𝑅𝑔)) = (𝑓𝐴, 𝑔𝐵 ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
174, 6, 163eqtr4i 2086 1 ( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓𝐴, 𝑔𝐵 ↦ (𝑓𝑓 𝑅𝑔))
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  cin 2943  wss 2944  cmpt 3845   × cxp 4370  dom cdm 4372  cres 4374  cfv 4929  (class class class)co 5539  cmpt2 5541  𝑓 cof 5737
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-in1 554  ax-in2 555  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-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-of 5739
This theorem is referenced by: (None)
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