Theorem offeq 6114

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 6113	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈)
8	7	ffnd 5381	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) Fn 𝐶)
9		offeq.4	. . 3 ⊢ (𝜑 → 𝐻:𝐶⟶𝑈)
10	9	ffnd 5381	. 2 ⊢ (𝜑 → 𝐻 Fn 𝐶)
11	2	ffnd 5381	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
12	3	ffnd 5381	. . . 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 5667	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝑈)
17	15, 16	eqeltrd 2266	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐷𝑅𝐸) ∈ 𝑈)
18	11, 12, 4, 5, 6, 13, 14, 17	ofvalg 6110	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐷𝑅𝐸))
19	18, 15	eqtrd 2222	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐻‘𝑥))
20	8, 10, 19	eqfnfvd 5632	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160   ∩ cin 3143  ⟶wf 5227  ‘cfv 5231  (class class class)co 5891   ∘_𝑓 cof 6099

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-setind 4551

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-of 6101

This theorem is referenced by:  dviaddf  14566