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Theorem ofresid 30317
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 6843 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
3 ofresid.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
43ffvelrnda 6843 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐵)
52, 4opelxpd 5586 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐵))
65fvresd 6683 . . . . 5 ((𝜑𝑥𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩))
76eqcomd 2824 . . . 4 ((𝜑𝑥𝐴) → (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩))
8 df-ov 7148 . . . 4 ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩)
9 df-ov 7148 . . . 4 ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩)
107, 8, 93eqtr4g 2878 . . 3 ((𝜑𝑥𝐴) → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥)))
1110mpteq2dva 5152 . 2 (𝜑 → (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝐴 ↦ ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥))))
121ffnd 6508 . . 3 (𝜑𝐹 Fn 𝐴)
133ffnd 6508 . . 3 (𝜑𝐺 Fn 𝐴)
14 ofresid.3 . . 3 (𝜑𝐴𝑉)
15 inidm 4192 . . 3 (𝐴𝐴) = 𝐴
16 eqidd 2819 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
17 eqidd 2819 . . 3 ((𝜑𝑥𝐴) → (𝐺𝑥) = (𝐺𝑥))
1812, 13, 14, 14, 15, 16, 17offval 7405 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
1912, 13, 14, 14, 15, 16, 17offval 7405 . 2 (𝜑 → (𝐹f (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥))))
2011, 18, 193eqtr4d 2863 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝐹f (𝑅 ↾ (𝐵 × 𝐵))𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cop 4563  cmpt 5137   × cxp 5546  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398
This theorem is referenced by:  sitmcl  31508
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