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Theorem ofres 7552
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 2739 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2739 . . 3 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 7542 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐶 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
9 inss1 4162 . . . . 5 (𝐴𝐵) ⊆ 𝐴
105, 9eqsstrri 3956 . . . 4 𝐶𝐴
11 fnssres 6555 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹𝐶) Fn 𝐶)
121, 10, 11sylancl 586 . . 3 (𝜑 → (𝐹𝐶) Fn 𝐶)
13 inss2 4163 . . . . 5 (𝐴𝐵) ⊆ 𝐵
145, 13eqsstrri 3956 . . . 4 𝐶𝐵
15 fnssres 6555 . . . 4 ((𝐺 Fn 𝐵𝐶𝐵) → (𝐺𝐶) Fn 𝐶)
162, 14, 15sylancl 586 . . 3 (𝜑 → (𝐺𝐶) Fn 𝐶)
17 ssexg 5247 . . . 4 ((𝐶𝐴𝐴𝑉) → 𝐶 ∈ V)
1810, 3, 17sylancr 587 . . 3 (𝜑𝐶 ∈ V)
19 inidm 4152 . . 3 (𝐶𝐶) = 𝐶
20 fvres 6793 . . . 4 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
2120adantl 482 . . 3 ((𝜑𝑥𝐶) → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
22 fvres 6793 . . . 4 (𝑥𝐶 → ((𝐺𝐶)‘𝑥) = (𝐺𝑥))
2322adantl 482 . . 3 ((𝜑𝑥𝐶) → ((𝐺𝐶)‘𝑥) = (𝐺𝑥))
2412, 16, 18, 18, 19, 21, 23offval 7542 . 2 (𝜑 → ((𝐹𝐶) ∘f 𝑅(𝐺𝐶)) = (𝑥𝐶 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
258, 24eqtr4d 2781 1 (𝜑 → (𝐹f 𝑅𝐺) = ((𝐹𝐶) ∘f 𝑅(𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  cin 3886  wss 3887  cmpt 5157  cres 5591   Fn wfn 6428  cfv 6433  (class class class)co 7275  f cof 7531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533
This theorem is referenced by: (None)
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