Theorem ofresid 30079

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	ffvelrnda 6716	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
3		ofresid.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffvelrnda 6716	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵)
5	2, 4	opelxpd 5481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐵))
6	5	fvresd 6558	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
7	6	eqcomd 2801	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
8		df-ov 7019	. . . 4 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
9		df-ov 7019	. . . 4 ⊢ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
10	7, 8, 9	3eqtr4g 2856	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))
11	10	mpteq2dva 5055	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))))
12	1	ffnd 6383	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
13	3	ffnd 6383	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
14		ofresid.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
15		inidm 4115	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
16		eqidd 2796	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
17		eqidd 2796	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
18	12, 13, 14, 14, 15, 16, 17	offval 7274	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
19	12, 13, 14, 14, 15, 16, 17	offval 7274	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))))
20	11, 18, 19	3eqtr4d 2841	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝐹 ∘𝑓 (𝑅 ↾ (𝐵 × 𝐵))𝐺))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ⟨cop 4478   ↦ cmpt 5041   × cxp 5441   ↾ cres 5445  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016   ∘_𝑓 cof 7265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267

This theorem is referenced by:  sitmcl  31226