Theorem ofun 41879

Description: A function operation of unions of disjoint functions is a union of function operations. (Contributed by SN, 16-Jun-2024.)

Hypotheses

Ref	Expression
ofun.a	⊢ (𝜑 → 𝐴 Fn 𝑀)
ofun.b	⊢ (𝜑 → 𝐵 Fn 𝑀)
ofun.c	⊢ (𝜑 → 𝐶 Fn 𝑁)
ofun.d	⊢ (𝜑 → 𝐷 Fn 𝑁)
ofun.m	⊢ (𝜑 → 𝑀 ∈ 𝑉)
ofun.n	⊢ (𝜑 → 𝑁 ∈ 𝑊)
ofun.1	⊢ (𝜑 → (𝑀 ∩ 𝑁) = ∅)

Assertion

Ref	Expression
ofun	⊢ (𝜑 → ((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷)) = ((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷)))

Proof of Theorem ofun

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofun.a	. . . 4 ⊢ (𝜑 → 𝐴 Fn 𝑀)
2		ofun.c	. . . 4 ⊢ (𝜑 → 𝐶 Fn 𝑁)
3		ofun.1	. . . 4 ⊢ (𝜑 → (𝑀 ∩ 𝑁) = ∅)
4	1, 2, 3	fnund 6670	. . 3 ⊢ (𝜑 → (𝐴 ∪ 𝐶) Fn (𝑀 ∪ 𝑁))
5		ofun.b	. . . 4 ⊢ (𝜑 → 𝐵 Fn 𝑀)
6		ofun.d	. . . 4 ⊢ (𝜑 → 𝐷 Fn 𝑁)
7	5, 6, 3	fnund 6670	. . 3 ⊢ (𝜑 → (𝐵 ∪ 𝐷) Fn (𝑀 ∪ 𝑁))
8		ofun.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑉)
9		ofun.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊)
10	8, 9	unexd 7757	. . 3 ⊢ (𝜑 → (𝑀 ∪ 𝑁) ∈ V)
11		inidm 4217	. . 3 ⊢ ((𝑀 ∪ 𝑁) ∩ (𝑀 ∪ 𝑁)) = (𝑀 ∪ 𝑁)
12	4, 7, 10, 10, 11	offn 7698	. 2 ⊢ (𝜑 → ((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷)) Fn (𝑀 ∪ 𝑁))
13		inidm 4217	. . . 4 ⊢ (𝑀 ∩ 𝑀) = 𝑀
14	1, 5, 8, 8, 13	offn 7698	. . 3 ⊢ (𝜑 → (𝐴 ∘f 𝑅𝐵) Fn 𝑀)
15		inidm 4217	. . . 4 ⊢ (𝑁 ∩ 𝑁) = 𝑁
16	2, 6, 9, 9, 15	offn 7698	. . 3 ⊢ (𝜑 → (𝐶 ∘f 𝑅𝐷) Fn 𝑁)
17	14, 16, 3	fnund 6670	. 2 ⊢ (𝜑 → ((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷)) Fn (𝑀 ∪ 𝑁))
18		eqidd 2726	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀 ∪ 𝑁)) → ((𝐴 ∪ 𝐶)‘𝑥) = ((𝐴 ∪ 𝐶)‘𝑥))
19		eqidd 2726	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀 ∪ 𝑁)) → ((𝐵 ∪ 𝐷)‘𝑥) = ((𝐵 ∪ 𝐷)‘𝑥))
20	4, 7, 10, 10, 11, 18, 19	ofval 7696	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀 ∪ 𝑁)) → (((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷))‘𝑥) = (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)))
21		elun 4145	. . . 4 ⊢ (𝑥 ∈ (𝑀 ∪ 𝑁) ↔ (𝑥 ∈ 𝑀 ∨ 𝑥 ∈ 𝑁))
22		eqidd 2726	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝐴‘𝑥) = (𝐴‘𝑥))
23		eqidd 2726	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝐵‘𝑥) = (𝐵‘𝑥))
24	1, 5, 8, 8, 13, 22, 23	ofval 7696	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝐴 ∘f 𝑅𝐵)‘𝑥) = ((𝐴‘𝑥)𝑅(𝐵‘𝑥)))
25	14	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝐴 ∘f 𝑅𝐵) Fn 𝑀)
26	16	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝐶 ∘f 𝑅𝐷) Fn 𝑁)
27	3	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝑀 ∩ 𝑁) = ∅)
28		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀)
29	25, 26, 27, 28	fvun1d 6990	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥) = ((𝐴 ∘f 𝑅𝐵)‘𝑥))
30	1	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 Fn 𝑀)
31	2	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐶 Fn 𝑁)
32	30, 31, 27, 28	fvun1d 6990	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝐴 ∪ 𝐶)‘𝑥) = (𝐴‘𝑥))
33	5	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐵 Fn 𝑀)
34	6	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐷 Fn 𝑁)
35	33, 34, 27, 28	fvun1d 6990	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝐵 ∪ 𝐷)‘𝑥) = (𝐵‘𝑥))
36	32, 35	oveq12d 7437	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = ((𝐴‘𝑥)𝑅(𝐵‘𝑥)))
37	24, 29, 36	3eqtr4rd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥))
38		eqidd 2726	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐶‘𝑥) = (𝐶‘𝑥))
39		eqidd 2726	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐷‘𝑥) = (𝐷‘𝑥))
40	2, 6, 9, 9, 15, 38, 39	ofval 7696	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → ((𝐶 ∘f 𝑅𝐷)‘𝑥) = ((𝐶‘𝑥)𝑅(𝐷‘𝑥)))
41	14	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐴 ∘f 𝑅𝐵) Fn 𝑀)
42	16	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐶 ∘f 𝑅𝐷) Fn 𝑁)
43	3	adantr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝑀 ∩ 𝑁) = ∅)
44		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ 𝑁)
45	41, 42, 43, 44	fvun2d 6991	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥) = ((𝐶 ∘f 𝑅𝐷)‘𝑥))
46	1	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐴 Fn 𝑀)
47	2	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐶 Fn 𝑁)
48	46, 47, 43, 44	fvun2d 6991	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → ((𝐴 ∪ 𝐶)‘𝑥) = (𝐶‘𝑥))
49	5	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐵 Fn 𝑀)
50	6	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐷 Fn 𝑁)
51	49, 50, 43, 44	fvun2d 6991	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → ((𝐵 ∪ 𝐷)‘𝑥) = (𝐷‘𝑥))
52	48, 51	oveq12d 7437	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = ((𝐶‘𝑥)𝑅(𝐷‘𝑥)))
53	40, 45, 52	3eqtr4rd 2776	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥))
54	37, 53	jaodan 955	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑀 ∨ 𝑥 ∈ 𝑁)) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥))
55	21, 54	sylan2b 592	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀 ∪ 𝑁)) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥))
56	20, 55	eqtrd 2765	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀 ∪ 𝑁)) → (((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷))‘𝑥) = (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥))
57	12, 17, 56	eqfnfvd 7042	1 ⊢ (𝜑 → ((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷)) = ((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷)))

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ∪ cun 3942   ∩ cin 3943  ∅c0 4322   Fn wfn 6544  ‘cfv 6549  (class class class)co 7419   ∘_f cof 7683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685

This theorem is referenced by:  fsuppssind  41980