Theorem br1cossxrncnvepres 37928

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 37924	. 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 37741	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢))
3	2	elv 3477	. . . . 5 ⊢ (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢)
4	3	anbi1i 622	. . . 4 ⊢ ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ↔ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵))
5		brcnvep 37741	. . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐸 ↔ 𝐸 ∈ 𝑢))
6	5	elv 3477	. . . . 5 ⊢ (𝑢◡ E 𝐸 ↔ 𝐸 ∈ 𝑢)
7	6	anbi1i 622	. . . 4 ⊢ ((𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷) ↔ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷))
8	4, 7	anbi12i 626	. . 3 ⊢ (((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷)))
9	8	rexbii 3090	. 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐸 ∧ 𝑢𝑅𝐷)) ↔ ∃𝑢 ∈ 𝐴 ((𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐸 ∈ 𝑢 ∧ 𝑢𝑅𝐷)))
10	1, 9	bitrdi 286	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 205   ∧ wa 394   ∈ wcel 2098  ∃wrex 3066  Vcvv 3471  ⟨cop 4636   class class class wbr 5150   E cep 5583  ^◡ccnv 5679   ↾ cres 5682   ⋉ cxrn 37652   ≀ ccoss 37653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-eprel 5584  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fo 6557  df-fv 6559  df-1st 7997  df-2nd 7998  df-xrn 37847  df-coss 37887

This theorem is referenced by: (None)

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