Theorem offeq 6244

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 6243	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈)
8	7	ffnd 5480	. 2 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) Fn 𝐶)
9		offeq.4	. . 3 ⊢ (𝜑 → 𝐻:𝐶⟶𝑈)
10	9	ffnd 5480	. 2 ⊢ (𝜑 → 𝐻 Fn 𝐶)
11	2	ffnd 5480	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
12	3	ffnd 5480	. . . 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 5778	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐻‘𝑥) ∈ 𝑈)
17	15, 16	eqeltrd 2306	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐷𝑅𝐸) ∈ 𝑈)
18	11, 12, 4, 5, 6, 13, 14, 17	ofvalg 6240	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐷𝑅𝐸))
19	18, 15	eqtrd 2262	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐻‘𝑥))
20	8, 10, 19	eqfnfvd 5743	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200   ∩ cin 3197  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013   ∘_𝑓 cof 6228

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230

This theorem is referenced by:  ofnegsub  9132  dviaddf  15419