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Theorem ofresid 32652
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)
21ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
3 ofresid.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
43ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐵)
52, 4opelxpd 5724 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐵))
65fvresd 6926 . . . . 5 ((𝜑𝑥𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩))
76eqcomd 2743 . . . 4 ((𝜑𝑥𝐴) → (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩))
8 df-ov 7434 . . . 4 ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩)
9 df-ov 7434 . . . 4 ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩)
107, 8, 93eqtr4g 2802 . . 3 ((𝜑𝑥𝐴) → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥)))
1110mpteq2dva 5242 . 2 (𝜑 → (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝐴 ↦ ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥))))
121ffnd 6737 . . 3 (𝜑𝐹 Fn 𝐴)
133ffnd 6737 . . 3 (𝜑𝐺 Fn 𝐴)
14 ofresid.3 . . 3 (𝜑𝐴𝑉)
15 inidm 4227 . . 3 (𝐴𝐴) = 𝐴
16 eqidd 2738 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
17 eqidd 2738 . . 3 ((𝜑𝑥𝐴) → (𝐺𝑥) = (𝐺𝑥))
1812, 13, 14, 14, 15, 16, 17offval 7706 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
1912, 13, 14, 14, 15, 16, 17offval 7706 . 2 (𝜑 → (𝐹f (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥))))
2011, 18, 193eqtr4d 2787 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝐹f (𝑅 ↾ (𝐵 × 𝐵))𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cop 4632  cmpt 5225   × cxp 5683  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by:  sitmcl  34353
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