Theorem ofresid 32616

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 7012	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
3		ofresid.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffvelcdmda 7012	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵)
5	2, 4	opelxpd 5650	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐵))
6	5	fvresd 6837	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
7	6	eqcomd 2737	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
8		df-ov 7344	. . . 4 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
9		df-ov 7344	. . . 4 ⊢ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
10	7, 8, 9	3eqtr4g 2791	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))
11	10	mpteq2dva 5179	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))))
12	1	ffnd 6647	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
13	3	ffnd 6647	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
14		ofresid.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
15		inidm 4172	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
16		eqidd 2732	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
17		eqidd 2732	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
18	12, 13, 14, 14, 15, 16, 17	offval 7614	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
19	12, 13, 14, 14, 15, 16, 17	offval 7614	. 2 ⊢ (𝜑 → (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))))
20	11, 18, 19	3eqtr4d 2776	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟨cop 4577   ↦ cmpt 5167   × cxp 5609   ↾ cres 5613  ⟶wf 6472  ‘cfv 6476  (class class class)co 7341   ∘_f cof 7603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605

This theorem is referenced by:  sitmcl  34356