Theorem ofresid 32658

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	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
ofresid.2	⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
ofresid.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
ofresid	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺))

Proof of Theorem ofresid

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofresid.1	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffvelcdmda 7103	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
3		ofresid.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffvelcdmda 7103	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵)
5	2, 4	opelxpd 5727	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐵))
6	5	fvresd 6926	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
7	6	eqcomd 2740	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
8		df-ov 7433	. . . 4 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
9		df-ov 7433	. . . 4 ⊢ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
10	7, 8, 9	3eqtr4g 2799	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))
11	10	mpteq2dva 5247	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))))
12	1	ffnd 6737	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
13	3	ffnd 6737	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
14		ofresid.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
15		inidm 4234	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
16		eqidd 2735	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
17		eqidd 2735	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
18	12, 13, 14, 14, 15, 16, 17	offval 7705	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
19	12, 13, 14, 14, 15, 16, 17	offval 7705	. 2 ⊢ (𝜑 → (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))))
20	11, 18, 19	3eqtr4d 2784	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ⟨cop 4636   ↦ cmpt 5230   × cxp 5686   ↾ cres 5690  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430   ∘_f cof 7694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696

This theorem is referenced by:  sitmcl  34332