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Theorem ofres 5964
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 (𝜑𝐴𝑉)
ofres.4 (𝜑𝐵𝑊)
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 (𝜑𝐴𝑉)
4 ofres.4 . . 3 (𝜑𝐵𝑊)
5 ofres.5 . . 3 (𝐴𝐵) = 𝐶
6 eqidd 2118 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2118 . . 3 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 5957 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐶 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
9 inss1 3266 . . . . 5 (𝐴𝐵) ⊆ 𝐴
105, 9eqsstrri 3100 . . . 4 𝐶𝐴
11 fnssres 5206 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹𝐶) Fn 𝐶)
121, 10, 11sylancl 409 . . 3 (𝜑 → (𝐹𝐶) Fn 𝐶)
13 inss2 3267 . . . . 5 (𝐴𝐵) ⊆ 𝐵
145, 13eqsstrri 3100 . . . 4 𝐶𝐵
15 fnssres 5206 . . . 4 ((𝐺 Fn 𝐵𝐶𝐵) → (𝐺𝐶) Fn 𝐶)
162, 14, 15sylancl 409 . . 3 (𝜑 → (𝐺𝐶) Fn 𝐶)
17 ssexg 4037 . . . 4 ((𝐶𝐴𝐴𝑉) → 𝐶 ∈ V)
1810, 3, 17sylancr 410 . . 3 (𝜑𝐶 ∈ V)
19 inidm 3255 . . 3 (𝐶𝐶) = 𝐶
20 fvres 5413 . . . 4 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
2120adantl 275 . . 3 ((𝜑𝑥𝐶) → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
22 fvres 5413 . . . 4 (𝑥𝐶 → ((𝐺𝐶)‘𝑥) = (𝐺𝑥))
2322adantl 275 . . 3 ((𝜑𝑥𝐶) → ((𝐺𝐶)‘𝑥) = (𝐺𝑥))
2412, 16, 18, 18, 19, 21, 23offval 5957 . 2 (𝜑 → ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)) = (𝑥𝐶 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
258, 24eqtr4d 2153 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  Vcvv 2660  cin 3040  wss 3041  cmpt 3959  cres 4511   Fn wfn 5088  cfv 5093  (class class class)co 5742  𝑓 cof 5948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-of 5950
This theorem is referenced by: (None)
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