Theorem br1cossxrnres 35219

Description: ⟨𝐵, 𝐶⟩ and ⟨𝐷, 𝐸⟩ are cosets by an intersection 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 35182	. . . 4 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
2	1	cosseqi 35203	. . 3 ⊢ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))
3	2	breqi 4968	. 2 ⊢ (⟨𝐵, 𝐶⟩ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))⟨𝐷, 𝐸⟩)
4		opex 5248	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
5		opex 5248	. . . 4 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
6		br1cossres 35215	. . . 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 688	. . 3 ⊢ (⟨𝐵, 𝐶⟩ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩))
8		brxrn 35157	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
9	8	el3v1 35024	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
10		brxrn 35157	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)))
11	10	el3v1 35024	. . . . . 6 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)))
12	9, 11	bi2anan9 635	. . . . 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 35030	. . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))
14	12, 13	syl6bb 288	. . . 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 3260	. . 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	syl5bb 284	. 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	syl5bbr 286	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 207   ∧ wa 396   ∈ wcel 2081  ∃wrex 3106  Vcvv 3437  ⟨cop 4478   class class class wbr 4962   ↾ cres 5445   ⋉ cxrn 34984   ≀ ccoss 34985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fo 6231  df-fv 6233  df-1st 7545  df-2nd 7546  df-xrn 35154  df-coss 35190

This theorem is referenced by:  br1cossxrnidres  35222  br1cossxrncnvepres  35223  br1cossxrncnvssrres  35279