Theorem br1cossxrnres 37313

Description: ⟨𝐵, 𝐶⟩ and ⟨𝐷, 𝐸⟩ are cosets by an range Cartesian product with a restriction: a binary relation. (Contributed by Peter Mazsa, 8-Jun-2021.)

Assertion

Ref	Expression
br1cossxrnres	⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐵   𝑢,𝐶   𝑢,𝐷   𝑢,𝐸   𝑢,𝑅   𝑢,𝑆   𝑢,𝑉   𝑢,𝑊   𝑢,𝑋   𝑢,𝑌

Proof of Theorem br1cossxrnres

Step	Hyp	Ref	Expression
1		xrnres2 37268	. . . 4 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
2	1	cosseqi 37292	. . 3 ⊢ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))
3	2	breqi 5154	. 2 ⊢ (⟨𝐵, 𝐶⟩ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))⟨𝐷, 𝐸⟩)
4		opex 5464	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
5		opex 5464	. . . 4 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
6		br1cossres 37304	. . . 4 ⊢ ((⟨𝐵, 𝐶⟩ ∈ V ∧ ⟨𝐷, 𝐸⟩ ∈ V) → (⟨𝐵, 𝐶⟩ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩)))
7	4, 5, 6	mp2an 690	. . 3 ⊢ (⟨𝐵, 𝐶⟩ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩))
8		brxrn 37239	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
9	8	el3v1 37082	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
10		brxrn 37239	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)))
11	10	el3v1 37082	. . . . . 6 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)))
12	9, 11	bi2anan9 637	. . . . 5 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → ((𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩) ↔ ((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸))))
13		an2anr 635	. . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))
14	12, 13	bitrdi 286	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → ((𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩) ↔ ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))))
15	14	rexbidv 3178	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩) ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))))
16	7, 15	bitrid 282	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))))
17	3, 16	bitr3id 284	1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∃wrex 3070  Vcvv 3474  ⟨cop 4634   class class class wbr 5148   ↾ cres 5678   ⋉ cxrn 37037   ≀ ccoss 37038

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7974  df-2nd 7975  df-xrn 37236  df-coss 37276

This theorem is referenced by:  br1cossxrnidres  37316  br1cossxrncnvepres  37317  br1cossxrncnvssrres  37373