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Theorem ofresid 30407
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 6832 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
3 ofresid.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
43ffvelrnda 6832 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐺𝑥) ∈ 𝐵)
52, 4opelxpd 5561 . . . . . 6 ((𝜑𝑥𝐴) → ⟨(𝐹𝑥), (𝐺𝑥)⟩ ∈ (𝐵 × 𝐵))
65fvresd 6669 . . . . 5 ((𝜑𝑥𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩) = (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩))
76eqcomd 2807 . . . 4 ((𝜑𝑥𝐴) → (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩))
8 df-ov 7142 . . . 4 ((𝐹𝑥)𝑅(𝐺𝑥)) = (𝑅‘⟨(𝐹𝑥), (𝐺𝑥)⟩)
9 df-ov 7142 . . . 4 ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹𝑥), (𝐺𝑥)⟩)
107, 8, 93eqtr4g 2861 . . 3 ((𝜑𝑥𝐴) → ((𝐹𝑥)𝑅(𝐺𝑥)) = ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥)))
1110mpteq2dva 5128 . 2 (𝜑 → (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))) = (𝑥𝐴 ↦ ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥))))
121ffnd 6492 . . 3 (𝜑𝐹 Fn 𝐴)
133ffnd 6492 . . 3 (𝜑𝐺 Fn 𝐴)
14 ofresid.3 . . 3 (𝜑𝐴𝑉)
15 inidm 4148 . . 3 (𝐴𝐴) = 𝐴
16 eqidd 2802 . . 3 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
17 eqidd 2802 . . 3 ((𝜑𝑥𝐴) → (𝐺𝑥) = (𝐺𝑥))
1812, 13, 14, 14, 15, 16, 17offval 7400 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
1912, 13, 14, 14, 15, 16, 17offval 7400 . 2 (𝜑 → (𝐹f (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺𝑥))))
2011, 18, 193eqtr4d 2846 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝐹f (𝑅 ↾ (𝐵 × 𝐵))𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  cop 4534  cmpt 5113   × cxp 5521  cres 5525  wf 6324  cfv 6328  (class class class)co 7139  f cof 7391
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393
This theorem is referenced by:  sitmcl  31723
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