Theorem br1cossxrncnvepres 35694

Description: ⟨𝐵, 𝐶⟩ and ⟨𝐷, 𝐸⟩ are cosets by a range Cartesian product with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 12-May-2021.)

Assertion

Ref	Expression
br1cossxrncnvepres	⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷))))

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

Proof of Theorem br1cossxrncnvepres

Step	Hyp	Ref	Expression
1		br1cossxrnres 35690	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷))))
2		brcnvep 35528	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢))
3	2	elv 3501	. . . . 5 ⊢ (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢)
4	3	anbi1i 625	. . . 4 ⊢ ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ↔ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵))
5		brcnvep 35528	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐸 ↔ 𝐸 ∈ 𝑢))
6	5	elv 3501	. . . . 5 ⊢ (𝑢◡ E 𝐸 ↔ 𝐸 ∈ 𝑢)
7	6	anbi1i 625	. . . 4 ⊢ ((𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷) ↔ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷))
8	4, 7	anbi12i 628	. . 3 ⊢ (((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷)))
9	8	rexbii 3249	. 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷)))
10	1, 9	syl6bb 289	1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) ∧ (𝐷 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌)) → (⟨𝐵, 𝐶⟩ ≀ (𝑅 ⋉ (◡ E ↾ 𝐴))⟨𝐷, 𝐸⟩ ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2114  ∃wrex 3141  Vcvv 3496  ⟨cop 4575   class class class wbr 5068   E cep 5466  ^◡ccnv 5556   ↾ cres 5559   ⋉ cxrn 35454   ≀ ccoss 35455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-eprel 5467  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fo 6363  df-fv 6365  df-1st 7691  df-2nd 7692  df-xrn 35625  df-coss 35661

This theorem is referenced by: (None)