Theorem ofun 42602

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 6615	. . 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 6615	. . 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 7709	. . 3 ⊢ (𝜑 → (𝑀 ∪ 𝑁) ∈ V)
11		inidm 4181	. . 3 ⊢ ((𝑀 ∪ 𝑁) ∩ (𝑀 ∪ 𝑁)) = (𝑀 ∪ 𝑁)
12	4, 7, 10, 10, 11	offn 7645	. 2 ⊢ (𝜑 → ((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷)) Fn (𝑀 ∪ 𝑁))
13		inidm 4181	. . . 4 ⊢ (𝑀 ∩ 𝑀) = 𝑀
14	1, 5, 8, 8, 13	offn 7645	. . 3 ⊢ (𝜑 → (𝐴 ∘f 𝑅𝐵) Fn 𝑀)
15		inidm 4181	. . . 4 ⊢ (𝑁 ∩ 𝑁) = 𝑁
16	2, 6, 9, 9, 15	offn 7645	. . 3 ⊢ (𝜑 → (𝐶 ∘f 𝑅𝐷) Fn 𝑁)
17	14, 16, 3	fnund 6615	. 2 ⊢ (𝜑 → ((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷)) Fn (𝑀 ∪ 𝑁))
18		eqidd 2738	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀 ∪ 𝑁)) → ((𝐴 ∪ 𝐶)‘𝑥) = ((𝐴 ∪ 𝐶)‘𝑥))
19		eqidd 2738	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀 ∪ 𝑁)) → ((𝐵 ∪ 𝐷)‘𝑥) = ((𝐵 ∪ 𝐷)‘𝑥))
20	4, 7, 10, 10, 11, 18, 19	ofval 7643	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀 ∪ 𝑁)) → (((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷))‘𝑥) = (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)))
21		elun 4107	. . . 4 ⊢ (𝑥 ∈ (𝑀 ∪ 𝑁) ↔ (𝑥 ∈ 𝑀 ∨ 𝑥 ∈ 𝑁))
22		eqidd 2738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝐴‘𝑥) = (𝐴‘𝑥))
23		eqidd 2738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝐵‘𝑥) = (𝐵‘𝑥))
24	1, 5, 8, 8, 13, 22, 23	ofval 7643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝐴 ∘f 𝑅𝐵)‘𝑥) = ((𝐴‘𝑥)𝑅(𝐵‘𝑥)))
25	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝐴 ∘f 𝑅𝐵) Fn 𝑀)
26	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝐶 ∘f 𝑅𝐷) Fn 𝑁)
27	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (𝑀 ∩ 𝑁) = ∅)
28		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝑥 ∈ 𝑀)
29	25, 26, 27, 28	fvun1d 6935	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥) = ((𝐴 ∘f 𝑅𝐵)‘𝑥))
30	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐴 Fn 𝑀)
31	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐶 Fn 𝑁)
32	30, 31, 27, 28	fvun1d 6935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝐴 ∪ 𝐶)‘𝑥) = (𝐴‘𝑥))
33	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐵 Fn 𝑀)
34	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → 𝐷 Fn 𝑁)
35	33, 34, 27, 28	fvun1d 6935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → ((𝐵 ∪ 𝐷)‘𝑥) = (𝐵‘𝑥))
36	32, 35	oveq12d 7386	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = ((𝐴‘𝑥)𝑅(𝐵‘𝑥)))
37	24, 29, 36	3eqtr4rd 2783	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑀) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥))
38		eqidd 2738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐶‘𝑥) = (𝐶‘𝑥))
39		eqidd 2738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐷‘𝑥) = (𝐷‘𝑥))
40	2, 6, 9, 9, 15, 38, 39	ofval 7643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → ((𝐶 ∘f 𝑅𝐷)‘𝑥) = ((𝐶‘𝑥)𝑅(𝐷‘𝑥)))
41	14	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐴 ∘f 𝑅𝐵) Fn 𝑀)
42	16	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝐶 ∘f 𝑅𝐷) Fn 𝑁)
43	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (𝑀 ∩ 𝑁) = ∅)
44		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝑥 ∈ 𝑁)
45	41, 42, 43, 44	fvun2d 6936	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥) = ((𝐶 ∘f 𝑅𝐷)‘𝑥))
46	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐴 Fn 𝑀)
47	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐶 Fn 𝑁)
48	46, 47, 43, 44	fvun2d 6936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → ((𝐴 ∪ 𝐶)‘𝑥) = (𝐶‘𝑥))
49	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐵 Fn 𝑀)
50	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → 𝐷 Fn 𝑁)
51	49, 50, 43, 44	fvun2d 6936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → ((𝐵 ∪ 𝐷)‘𝑥) = (𝐷‘𝑥))
52	48, 51	oveq12d 7386	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = ((𝐶‘𝑥)𝑅(𝐷‘𝑥)))
53	40, 45, 52	3eqtr4rd 2783	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑁) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥))
54	37, 53	jaodan 960	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑀 ∨ 𝑥 ∈ 𝑁)) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥))
55	21, 54	sylan2b 595	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀 ∪ 𝑁)) → (((𝐴 ∪ 𝐶)‘𝑥)𝑅((𝐵 ∪ 𝐷)‘𝑥)) = (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥))
56	20, 55	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀 ∪ 𝑁)) → (((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷))‘𝑥) = (((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷))‘𝑥))
57	12, 17, 56	eqfnfvd 6988	1 ⊢ (𝜑 → ((𝐴 ∪ 𝐶) ∘f 𝑅(𝐵 ∪ 𝐷)) = ((𝐴 ∘f 𝑅𝐵) ∪ (𝐶 ∘f 𝑅𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∪ cun 3901   ∩ cin 3902  ∅c0 4287   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368   ∘_f cof 7630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632

This theorem is referenced by:  fsuppssind  42945