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Theorem ofres 5667
Description: Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
Hypotheses
Ref Expression
ofres.1 (φ𝐹 Fn A)
ofres.2 (φ𝐺 Fn B)
ofres.3 (φA 𝑉)
ofres.4 (φB 𝑊)
ofres.5 (AB) = 𝐶
Assertion
Ref Expression
ofres (φ → (𝐹𝑓 𝑅𝐺) = ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)))

Proof of Theorem ofres
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 ofres.1 . . 3 (φ𝐹 Fn A)
2 ofres.2 . . 3 (φ𝐺 Fn B)
3 ofres.3 . . 3 (φA 𝑉)
4 ofres.4 . . 3 (φB 𝑊)
5 ofres.5 . . 3 (AB) = 𝐶
6 eqidd 2038 . . 3 ((φ x A) → (𝐹x) = (𝐹x))
7 eqidd 2038 . . 3 ((φ x B) → (𝐺x) = (𝐺x))
81, 2, 3, 4, 5, 6, 7offval 5661 . 2 (φ → (𝐹𝑓 𝑅𝐺) = (x 𝐶 ↦ ((𝐹x)𝑅(𝐺x))))
9 inss1 3151 . . . . 5 (AB) ⊆ A
105, 9eqsstr3i 2970 . . . 4 𝐶A
11 fnssres 4955 . . . 4 ((𝐹 Fn A 𝐶A) → (𝐹𝐶) Fn 𝐶)
121, 10, 11sylancl 392 . . 3 (φ → (𝐹𝐶) Fn 𝐶)
13 inss2 3152 . . . . 5 (AB) ⊆ B
145, 13eqsstr3i 2970 . . . 4 𝐶B
15 fnssres 4955 . . . 4 ((𝐺 Fn B 𝐶B) → (𝐺𝐶) Fn 𝐶)
162, 14, 15sylancl 392 . . 3 (φ → (𝐺𝐶) Fn 𝐶)
17 ssexg 3887 . . . 4 ((𝐶A A 𝑉) → 𝐶 V)
1810, 3, 17sylancr 393 . . 3 (φ𝐶 V)
19 inidm 3140 . . 3 (𝐶𝐶) = 𝐶
20 fvres 5141 . . . 4 (x 𝐶 → ((𝐹𝐶)‘x) = (𝐹x))
2120adantl 262 . . 3 ((φ x 𝐶) → ((𝐹𝐶)‘x) = (𝐹x))
22 fvres 5141 . . . 4 (x 𝐶 → ((𝐺𝐶)‘x) = (𝐺x))
2322adantl 262 . . 3 ((φ x 𝐶) → ((𝐺𝐶)‘x) = (𝐺x))
2412, 16, 18, 18, 19, 21, 23offval 5661 . 2 (φ → ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)) = (x 𝐶 ↦ ((𝐹x)𝑅(𝐺x))))
258, 24eqtr4d 2072 1 (φ → (𝐹𝑓 𝑅𝐺) = ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  Vcvv 2551  cin 2910  wss 2911  cmpt 3809  cres 4290   Fn wfn 4840  cfv 4845  (class class class)co 5455  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-bndl 1396  ax-4 1397  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-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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