Theorem br1cossxrnres 38877

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 38765	. . . 4 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
2	1	cosseqi 38856	. . 3 ⊢ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))
3	2	breqi 5092	. 2 ⊢ (⟨𝐵, 𝐶⟩ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))⟨𝐷, 𝐸⟩)
4		opex 5413	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
5		opex 5413	. . . 4 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
6		br1cossres 38868	. . . 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 693	. . 3 ⊢ (⟨𝐵, 𝐶⟩ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩))
8		brxrn 38722	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
9	8	el3v1 38569	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
10		brxrn 38722	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)))
11	10	el3v1 38569	. . . . . 6 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)))
12	9, 11	bi2anan9 639	. . . . 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 637	. . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))
14	12, 13	bitrdi 287	. . . 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 3162	. . 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 283	. 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 285	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 206   ∧ wa 395   ∈ wcel 2114  ∃wrex 3062  Vcvv 3430  ⟨cop 4574   class class class wbr 5086   ↾ cres 5628   ⋉ cxrn 38513   ≀ ccoss 38522

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-sep 5232  ax-nul 5242  ax-pr 5372  ax-un 7684

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-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-fo 6500  df-fv 6502  df-1st 7937  df-2nd 7938  df-xrn 38719  df-coss 38840

This theorem is referenced by:  br1cossxrnidres  38880  br1cossxrncnvepres  38881  br1cossxrncnvssrres  38927