Theorem offeq 6063

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 6062	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈)
8	7	ffnd 5338	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) Fn 𝐶)
9		offeq.4	. . 3 ⊢ (𝜑 → 𝐻:𝐶⟶𝑈)
10	9	ffnd 5338	. 2 ⊢ (𝜑 → 𝐻 Fn 𝐶)
11	2	ffnd 5338	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
12	3	ffnd 5338	. . . 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	ffvelrnda 5620	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝑈)
17	15, 16	eqeltrd 2243	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐷𝑅𝐸) ∈ 𝑈)
18	11, 12, 4, 5, 6, 13, 14, 17	ofvalg 6059	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐷𝑅𝐸))
19	18, 15	eqtrd 2198	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐻‘𝑥))
20	8, 10, 19	eqfnfvd 5586	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136   ∩ cin 3115  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842   ∘_𝑓 cof 6048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050

This theorem is referenced by:  dviaddf  13309