Theorem br1cosscnvxrn 38497

Description: 𝐴 and 𝐵 are cosets by the converse range Cartesian product: a binary relation. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)

Assertion

Ref	Expression
br1cosscnvxrn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡(𝑅 ⋉ 𝑆)𝐵 ↔ (𝐴 ≀ ◡𝑅𝐵 ∧ 𝐴 ≀ ◡𝑆𝐵)))

Proof of Theorem br1cosscnvxrn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ecxrn 38410	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)})
2		ecxrn 38410	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → [𝐵](𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)})
3	1, 2	ineqan12d 4202	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)}))
4		inopab 5813	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))}
5	3, 4	eqtrdi 2787	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))})
6		an4 656	. . . . . 6 ⊢ (((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)) ↔ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
7	6	opabbii 5191	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))}
8	5, 7	eqtrdi 2787	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))})
9	8	neeq1d 2992	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))} ≠ ∅))
10		opabn0 5533	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))} ≠ ∅ ↔ ∃𝑥∃𝑦((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
11		exdistrv 1955	. . . 4 ⊢ (∃𝑥∃𝑦((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)) ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
12	10, 11	bitri 275	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))} ≠ ∅ ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
13	9, 12	bitrdi 287	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))))
14		brcosscnv2 38496	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡(𝑅 ⋉ 𝑆)𝐵 ↔ ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅))
15		brcosscnv 38495	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
16		brcosscnv 38495	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑆𝐵 ↔ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
17	15, 16	anbi12d 632	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ≀ ◡𝑅𝐵 ∧ 𝐴 ≀ ◡𝑆𝐵) ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))))
18	13, 14, 17	3bitr4d 311	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡(𝑅 ⋉ 𝑆)𝐵 ↔ (𝐴 ≀ ◡𝑅𝐵 ∧ 𝐴 ≀ ◡𝑆𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933   ∩ cin 3930  ∅c0 4313   class class class wbr 5124  {copab 5186  ^◡ccnv 5658  [cec 8722   ⋉ cxrn 38203   ≀ ccoss 38204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7993  df-2nd 7994  df-ec 8726  df-xrn 38394  df-coss 38434

This theorem is referenced by:  1cosscnvxrn  38498