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Theorem ofmres 5705
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 5706, allowing it to be used as a function or structure argument. By ofmresval 5665, 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)) = (f A, g B ↦ (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 2959 . . 3 A ⊆ V
2 ssv 2959 . . 3 B ⊆ V
3 resmpt2 5541 . . 3 ((A ⊆ V B ⊆ V) → ((f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx)))) ↾ (A × B)) = (f A, g B ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx)))))
41, 2, 3mp2an 402 . 2 ((f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx)))) ↾ (A × B)) = (f A, g B ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))))
5 df-of 5654 . . 3 𝑓 𝑅 = (f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))))
65reseq1i 4551 . 2 ( ∘𝑓 𝑅 ↾ (A × B)) = ((f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx)))) ↾ (A × B))
7 eqid 2037 . . 3 A = A
8 eqid 2037 . . 3 B = B
9 vex 2554 . . . 4 f V
10 vex 2554 . . . 4 g V
119dmex 4541 . . . . . 6 dom f V
1211inex1 3882 . . . . 5 (dom f ∩ dom g) V
1312mptex 5330 . . . 4 (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))) V
145ovmpt4g 5565 . . . 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 1231 . . 3 (f𝑓 𝑅g) = (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx)))
167, 8, 15mpt2eq123i 5510 . 2 (f A, g B ↦ (f𝑓 𝑅g)) = (f A, g B ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))))
174, 6, 163eqtr4i 2067 1 ( ∘𝑓 𝑅 ↾ (A × B)) = (f A, g B ↦ (f𝑓 𝑅g))
Colors of variables: wff set class
Syntax hints:   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  wss 2911  cmpt 3809   × cxp 4286  dom cdm 4288  cres 4290  cfv 4845  (class class class)co 5455  cmpt2 5457  𝑓 cof 5652
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654
This theorem is referenced by: (None)
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