Theorem ofresid 32646

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 7026	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
3		ofresid.2	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
4	3	ffvelcdmda 7026	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ 𝐵)
5	2, 4	opelxpd 5660	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩ ∈ (𝐵 × 𝐵))
6	5	fvresd 6851	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
7	6	eqcomd 2739	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩))
8		df-ov 7358	. . . 4 ⊢ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = (𝑅‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
9		df-ov 7358	. . . 4 ⊢ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)) = ((𝑅 ↾ (𝐵 × 𝐵))‘⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)
10	7, 8, 9	3eqtr4g 2793	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥)) = ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥)))
11	10	mpteq2dva 5188	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))))
12	1	ffnd 6660	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
13	3	ffnd 6660	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
14		ofresid.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
15		inidm 4176	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
16		eqidd 2734	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
17		eqidd 2734	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = (𝐺‘𝑥))
18	12, 13, 14, 14, 15, 16, 17	offval 7628	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
19	12, 13, 14, 14, 15, 16, 17	offval 7628	. 2 ⊢ (𝜑 → (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)(𝑅 ↾ (𝐵 × 𝐵))(𝐺‘𝑥))))
20	11, 18, 19	3eqtr4d 2778	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ⟨cop 4583   ↦ cmpt 5176   × cxp 5619   ↾ cres 5623  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   ∘_f cof 7617

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-of 7619

This theorem is referenced by:  sitmcl  34436