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Theorem ofres 7628
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 2738 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2738 . . 3 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 7618 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐶 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
9 inss1 4186 . . . . 5 (𝐴𝐵) ⊆ 𝐴
105, 9eqsstrri 3977 . . . 4 𝐶𝐴
11 fnssres 6621 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹𝐶) Fn 𝐶)
121, 10, 11sylancl 586 . . 3 (𝜑 → (𝐹𝐶) Fn 𝐶)
13 inss2 4187 . . . . 5 (𝐴𝐵) ⊆ 𝐵
145, 13eqsstrri 3977 . . . 4 𝐶𝐵
15 fnssres 6621 . . . 4 ((𝐺 Fn 𝐵𝐶𝐵) → (𝐺𝐶) Fn 𝐶)
162, 14, 15sylancl 586 . . 3 (𝜑 → (𝐺𝐶) Fn 𝐶)
17 ssexg 5278 . . . 4 ((𝐶𝐴𝐴𝑉) → 𝐶 ∈ V)
1810, 3, 17sylancr 587 . . 3 (𝜑𝐶 ∈ V)
19 inidm 4176 . . 3 (𝐶𝐶) = 𝐶
20 fvres 6858 . . . 4 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
2120adantl 482 . . 3 ((𝜑𝑥𝐶) → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
22 fvres 6858 . . . 4 (𝑥𝐶 → ((𝐺𝐶)‘𝑥) = (𝐺𝑥))
2322adantl 482 . . 3 ((𝜑𝑥𝐶) → ((𝐺𝐶)‘𝑥) = (𝐺𝑥))
2412, 16, 18, 18, 19, 21, 23offval 7618 . 2 (𝜑 → ((𝐹𝐶) ∘f 𝑅(𝐺𝐶)) = (𝑥𝐶 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
258, 24eqtr4d 2780 1 (𝜑 → (𝐹f 𝑅𝐺) = ((𝐹𝐶) ∘f 𝑅(𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3443  cin 3907  wss 3908  cmpt 5186  cres 5633   Fn wfn 6488  cfv 6493  (class class class)co 7351  f cof 7607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-of 7609
This theorem is referenced by:  ofoafg  41568
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