Theorem offeq 6149

Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)

Hypotheses

Ref	Expression
off.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
off.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
off.3	⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
off.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
off.5	⊢ (𝜑 → 𝐵 ∈ 𝑊)
off.6	⊢ (𝐴 ∩ 𝐵) = 𝐶
offeq.4	⊢ (𝜑 → 𝐻:𝐶⟶𝑈)
offeq.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐷)
offeq.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐸)
offeq.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐷𝑅𝐸) = (𝐻‘𝑥))

Assertion

Ref	Expression
offeq	⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)

Distinct variable groups:   𝑦,𝐺,𝑥   𝜑,𝑥,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦   𝑥,𝐻   𝑥,𝐺   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐻(𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem offeq

Step	Hyp	Ref	Expression
1		off.1	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
2		off.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
3		off.3	. . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
4		off.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		off.5	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6		off.6	. . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝐶
7	1, 2, 3, 4, 5, 6	off 6148	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈)
8	7	ffnd 5408	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) Fn 𝐶)
9		offeq.4	. . 3 ⊢ (𝜑 → 𝐻:𝐶⟶𝑈)
10	9	ffnd 5408	. 2 ⊢ (𝜑 → 𝐻 Fn 𝐶)
11	2	ffnd 5408	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
12	3	ffnd 5408	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
13		offeq.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐷)
14		offeq.6	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐸)
15		offeq.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐷𝑅𝐸) = (𝐻‘𝑥))
16	9	ffvelcdmda 5697	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝑈)
17	15, 16	eqeltrd 2273	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐷𝑅𝐸) ∈ 𝑈)
18	11, 12, 4, 5, 6, 13, 14, 17	ofvalg 6145	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐷𝑅𝐸))
19	18, 15	eqtrd 2229	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐻‘𝑥))
20	8, 10, 19	eqfnfvd 5662	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167   ∩ cin 3156  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922   ∘_𝑓 cof 6133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135

This theorem is referenced by:  ofnegsub  8989  dviaddf  14941