Theorem inxpxrn 39127

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 39091	. 2 ⊢ Rel ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))
2		relinxp 5803	. 2 ⊢ Rel ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))
3		brxrn2 39093	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢(𝑅 ⋉ 𝑆)𝑥 ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))
4	3	elv 3462	. . . . 5 ⊢ (𝑢(𝑅 ⋉ 𝑆)𝑥 ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))
5	4	anbi2i 635	. . . 4 ⊢ ((𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))
6	5	anbi2i 635	. . 3 ⊢ ((𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
7		xrninxp2 39125	. . . 4 ⊢ ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) = {⟨𝑢, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥))}
8	7	brabidgaw 39082	. . 3 ⊢ (𝑢((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))𝑥 ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ 𝑢(𝑅 ⋉ 𝑆)𝑥)))
9		brxrn2 39093	. . . . 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 3462	. . . 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 1111	. . . . 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 1882	. . . 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 5741	. . . . . . . . . . . 12 ⊢ (𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ↔ ((𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢𝑅𝑦))
14		brinxp2 5741	. . . . . . . . . . . 12 ⊢ (𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧 ↔ ((𝑢 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) ∧ 𝑢𝑆𝑧))
15	13, 14	anbi12i 640	. . . . . . . . . . 11 ⊢ ((𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ (((𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢𝑅𝑦) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) ∧ 𝑢𝑆𝑧)))
16		anan 38944	. . . . . . . . . . 11 ⊢ ((((𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢𝑅𝑦) ∧ ((𝑢 ∈ 𝐴 ∧ 𝑧 ∈ 𝐶) ∧ 𝑢𝑆𝑧)) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
17	15, 16	bitri 278	. . . . . . . . . 10 ⊢ ((𝑢(𝑅 ∩ (𝐴 × 𝐵))𝑦 ∧ 𝑢(𝑆 ∩ (𝐴 × 𝐶))𝑧) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
18	17	anbi2i 635	. . . . . . . . 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 474	. . . . . . . . 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 38950	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)))
21		opelxp 5699	. . . . . . . . . . . 12 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
22	21	anbi2i 635	. . . . . . . . . . 11 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ⟨𝑦, 𝑧⟩ ∈ (𝐵 × 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)))
23	20, 22	bitr2i 279	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)))
24	23	anbi1i 636	. . . . . . . . 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 302	. . . . . . . 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 466	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ↔ (𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩))
27	26	anbi1i 636	. . . . . . . 8 ⊢ (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑥 ∈ (𝐵 × 𝐶)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ ((𝑥 ∈ (𝐵 × 𝐶) ∧ 𝑥 = ⟨𝑦, 𝑧⟩) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
28		anass 474	. . . . . . . 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 300	. . . . . . 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 658	. . . . . . . . 9 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ (𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
31		3anass 1111	. . . . . . . . . 10 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))
32	31	anbi2i 635	. . . . . . . . 9 ⊢ ((𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)) ↔ (𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))))
33	30, 32	bitr4i 281	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑢 ∈ 𝐴 ∧ (𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧))) ↔ (𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))
34	33	anbi2i 635	. . . . . . 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 278	. . . . . 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 1882	. . . . 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 1990	. . . . 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 1990	. . . . . 6 ⊢ (∃𝑦∃𝑧(𝑢 ∈ 𝐴 ∧ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)) ↔ (𝑢 ∈ 𝐴 ∧ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑢𝑅𝑦 ∧ 𝑢𝑆𝑧)))
39	38	anbi2i 635	. . . . 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 300	. . . 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 300	. . 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 307	. 2 ⊢ (𝑢((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶)))𝑥 ↔ 𝑢((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))𝑥)
43	1, 2, 42	eqbrriv 5779	1 ⊢ ((𝑅 ∩ (𝐴 × 𝐵)) ⋉ (𝑆 ∩ (𝐴 × 𝐶))) = ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3457   ∩ cin 3905  ⟨cop 4597   class class class wbr 5111   × cxp 5661   ⋉ cxrn 38883

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7992  df-2nd 7993  df-xrn 39089

This theorem is used by:  xrnres4  39137  xrnresex  39138