Theorem inxpxrn 38371

Description: Two ways to express the intersection of a range Cartesian product with a Cartesian product. (Contributed by Peter Mazsa, 10-Apr-2020.)

Assertion

Ref	Expression
inxpxrn	⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))

Proof of Theorem inxpxrn

Dummy variables 𝑢 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xrnrel 38349	. 2 ⊢ Rel ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))
2		relinxp 5804	. 2 ⊢ Rel ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))
3		brxrn2 38351	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑥 ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))
4	3	elv 3468	. . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑥 ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))
5	4	anbi2i 623	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))
6	5	anbi2i 623	. . 3 ⊢ ((𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
7		xrninxp2 38369	. . . 4 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}
8	7	brabidgaw 38341	. . 3 ⊢ (𝑢((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))𝑥 ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)))
9		brxrn2 38351	. . . . 5 ⊢ (𝑢 ∈ V → (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)))
10	9	elv 3468	. . . 4 ⊢ (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧))
11		3anass 1094	. . . . 5 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)))
12	11	2exbii 1848	. . . 4 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)))
13		brinxp2 5743	. . . . . . . . . . . 12 ⊢ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢𝑅𝑦))
14		brinxp2 5743	. . . . . . . . . . . 12 ⊢ (𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧 ↔ ((𝑢 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) ∧ 𝑢𝑆𝑧))
15	13, 14	anbi12i 628	. . . . . . . . . . 11 ⊢ ((𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ (((𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢𝑅𝑦) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) ∧ 𝑢𝑆𝑧)))
16		anan 38205	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢𝑅𝑦) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) ∧ 𝑢𝑆𝑧)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
17	15, 16	bitri 275	. . . . . . . . . 10 ⊢ ((𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
18	17	anbi2i 623	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))))
19		anass 468	. . . . . . . . 9 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))))
20		eqelb 38211	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)))
21		opelxp 5701	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
22	21	anbi2i 623	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
23	20, 22	bitr2i 276	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)))
24	23	anbi1i 624	. . . . . . . . 9 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
25	18, 19, 24	3bitr2i 299	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
26		ancom 460	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩))
27	26	anbi1i 624	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ ((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
28		anass 468	. . . . . . . 8 ⊢ (((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))))
29	25, 27, 28	3bitri 297	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))))
30		an12 645	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ (𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
31		3anass 1094	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))
32	31	anbi2i 623	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)) ↔ (𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
33	30, 32	bitr4i 278	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ (𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))
34	33	anbi2i 623	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
35	29, 34	bitri 275	. . . . . 6 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
36	35	2exbii 1848	. . . . 5 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ ∃𝑦∃𝑧(𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
37		19.42vv 1956	. . . . 5 ⊢ (∃𝑦∃𝑧(𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ ∃𝑦∃𝑧(𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
38		19.42vv 1956	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))
39	38	anbi2i 623	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐶) ∧ ∃𝑦∃𝑧(𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
40	36, 37, 39	3bitri 297	. . . 4 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
41	10, 12, 40	3bitri 297	. . 3 ⊢ (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
42	6, 8, 41	3bitr4ri 304	. 2 ⊢ (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ 𝑢((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))𝑥)
43	1, 2, 42	eqbrriv 5781	1 ⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3463   ∩ cin 3930  ⟨cop 4612   class class class wbr 5123   × cxp 5663   ⋉ cxrn 38156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fo 6547  df-fv 6549  df-1st 7996  df-2nd 7997  df-xrn 38347

This theorem is referenced by:  xrnres4  38381  xrnresex  38382