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Theorem ofresid 28824
 Description: Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)
Hypotheses
Ref Expression
ofresid.1 (𝜑𝐹:𝐴𝐵)
ofresid.2 (𝜑𝐺:𝐴𝐵)
ofresid.3 (𝜑𝐴𝑉)
Assertion
Ref Expression
ofresid (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐹𝑓 (𝑅 ↾ (𝐵 × 𝐵))𝐺))

Proof of Theorem ofresid
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ofresid.1 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
21ffvelrnda 6267 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
3 ofresid.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
43ffvelrnda 6267 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐵)
5 opelxp 5070 . . . . . . 7 (⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐵) ↔ ((𝐹𝑥) ∈ 𝐵 ∧ (𝐺𝑥) ∈ 𝐵))
62, 4, 5sylanbrc 695 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐵))
7 fvres 6117 . . . . . 6 (⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐵) → ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩))
86, 7syl 17 . . . . 5 ((𝜑𝑥𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩))
98eqcomd 2616 . . . 4 ((𝜑𝑥𝐴) → (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩))
10 df-ov 6552 . . . 4 ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩)
11 df-ov 6552 . . . 4 ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩)
129, 10, 113eqtr4g 2669 . . 3 ((𝜑𝑥𝐴) → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥)))
1312mpteq2dva 4672 . 2 (𝜑 → (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝐴 ↦ ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥))))
14 ffn 5958 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
151, 14syl 17 . . 3 (𝜑𝐹 Fn 𝐴)
16 ffn 5958 . . . 4 (𝐺:𝐴𝐵𝐺 Fn 𝐴)
173, 16syl 17 . . 3 (𝜑𝐺 Fn 𝐴)
18 ofresid.3 . . 3 (𝜑𝐴𝑉)
19 inidm 3784 . . 3 (𝐴𝐴) = 𝐴
20 eqidd 2611 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
21 eqidd 2611 . . 3 ((𝜑𝑥𝐴) → (𝐺𝑥) = (𝐺𝑥))
2215, 17, 18, 18, 19, 20, 21offval 6802 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
2315, 17, 18, 18, 19, 20, 21offval 6802 . 2 (𝜑 → (𝐹𝑓 (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥))))
2413, 22, 233eqtr4d 2654 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐹𝑓 (𝑅 ↾ (𝐵 × 𝐵))𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795 This theorem is referenced by:  sitmcl  29740
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