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Theorem ofres 7079
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 2761 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2761 . . 3 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7offval 7070 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐶 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
9 inss1 3976 . . . . 5 (𝐴𝐵) ⊆ 𝐴
105, 9eqsstr3i 3777 . . . 4 𝐶𝐴
11 fnssres 6165 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → (𝐹𝐶) Fn 𝐶)
121, 10, 11sylancl 697 . . 3 (𝜑 → (𝐹𝐶) Fn 𝐶)
13 inss2 3977 . . . . 5 (𝐴𝐵) ⊆ 𝐵
145, 13eqsstr3i 3777 . . . 4 𝐶𝐵
15 fnssres 6165 . . . 4 ((𝐺 Fn 𝐵𝐶𝐵) → (𝐺𝐶) Fn 𝐶)
162, 14, 15sylancl 697 . . 3 (𝜑 → (𝐺𝐶) Fn 𝐶)
17 ssexg 4956 . . . 4 ((𝐶𝐴𝐴𝑉) → 𝐶 ∈ V)
1810, 3, 17sylancr 698 . . 3 (𝜑𝐶 ∈ V)
19 inidm 3965 . . 3 (𝐶𝐶) = 𝐶
20 fvres 6369 . . . 4 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
2120adantl 473 . . 3 ((𝜑𝑥𝐶) → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
22 fvres 6369 . . . 4 (𝑥𝐶 → ((𝐺𝐶)‘𝑥) = (𝐺𝑥))
2322adantl 473 . . 3 ((𝜑𝑥𝐶) → ((𝐺𝐶)‘𝑥) = (𝐺𝑥))
2412, 16, 18, 18, 19, 21, 23offval 7070 . 2 (𝜑 → ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)) = (𝑥𝐶 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
258, 24eqtr4d 2797 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = ((𝐹𝐶) ∘𝑓 𝑅(𝐺𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  wss 3715  cmpt 4881  cres 5268   Fn wfn 6044  cfv 6049  (class class class)co 6814  𝑓 cof 7061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063
This theorem is referenced by: (None)
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