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Theorem ofmres 5686
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 5687, allowing it to be used as a function or structure argument. By ofmresval 5646, 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 2942 . . 3 A ⊆ V
2 ssv 2942 . . 3 B ⊆ V
3 resmpt2 5522 . . 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 404 . 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 5635 . . 3 𝑓 𝑅 = (f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))))
65reseq1i 4535 . 2 ( ∘𝑓 𝑅 ↾ (A × B)) = ((f V, g V ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx)))) ↾ (A × B))
7 eqid 2022 . . 3 A = A
8 eqid 2022 . . 3 B = B
9 vex 2538 . . . 4 f V
10 vex 2538 . . . 4 g V
119dmex 4525 . . . . . 6 dom f V
1211inex1 3865 . . . . 5 (dom f ∩ dom g) V
1312mptex 5312 . . . 4 (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))) V
145ovmpt4g 5546 . . . 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 1217 . . 3 (f𝑓 𝑅g) = (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx)))
167, 8, 15mpt2eq123i 5491 . 2 (f A, g B ↦ (f𝑓 𝑅g)) = (f A, g B ↦ (x (dom f ∩ dom g) ↦ ((fx)𝑅(gx))))
174, 6, 163eqtr4i 2052 1 ( ∘𝑓 𝑅 ↾ (A × B)) = (f A, g B ↦ (f𝑓 𝑅g))
Colors of variables: wff set class
Syntax hints:   = wceq 1228   wcel 1374  Vcvv 2535  cin 2893  wss 2894  cmpt 3792   × cxp 4270  dom cdm 4272  cres 4274  cfv 4829  (class class class)co 5436  cmpt2 5438  𝑓 cof 5633
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-of 5635
This theorem is referenced by: (None)
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