Theorem br1cossxrnres 39661

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 39549	. . . 4 ⊢ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆 ↾ 𝐴))
2	1	cosseqi 39640	. . 3 ⊢ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴) = ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))
3	2	breqi 5112	. 2 ⊢ (⟨𝐵, 𝐶⟩ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))⟨𝐷, 𝐸⟩)
4		opex 5436	. . . 4 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
5		opex 5436	. . . 4 ⊢ ⟨𝐷, 𝐸⟩ ∈ V
6		br1cossres 39652	. . . 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 705	. . 3 ⊢ (⟨𝐵, 𝐶⟩ ≀ ((𝑅 ⋉ 𝑆) ↾ 𝐴)⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ∧ 𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩))
8		brxrn 39506	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
9	8	el3v1 39353	. . . . . 6 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐵, 𝐶⟩ ↔ (𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶)))
10		brxrn 39506	. . . . . . 7 ⊢ ((𝑢 ∈ V ∧ 𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)))
11	10	el3v1 39353	. . . . . 6 ⊢ ((𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑢(𝑅 ⋉ 𝑆)⟨𝐷, 𝐸⟩ ↔ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)))
12	9, 11	bi2anan9 650	. . . . 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 648	. . . . 5 ⊢ (((𝑢𝑅𝐵 ∧ 𝑢𝑆𝐶) ∧ (𝑢𝑅𝐷 ∧ 𝑢𝑆𝐸)) ↔ ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷)))
14	12, 13	bitrdi 290	. . . 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 3190	. . 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 286	. 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 288	1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (𝑆 ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢𝑆𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢𝑆𝐸 ∧ 𝑢𝑅𝐷))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454  ⟨cop 4838   class class class wbr 5106   ↾ cres 5657   ⋉ cxrn 39297   ≀ ccoss 39306

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fo 6561  df-fv 6563  df-1st 8022  df-2nd 8023  df-xrn 39503  df-coss 39624

This theorem is used by:  br1cossxrnidres  39664  br1cossxrncnvepres  39665  br1cossxrncnvssrres  39711