Theorem br1cosscnvxrn 38876

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 38718	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)})
2		ecxrn 38718	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → [𝐵](𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)})
3	1, 2	ineqan12d 4163	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)}))
4		inopab 5776	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))}
5	3, 4	eqtrdi 2788	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))})
6		an4 657	. . . . . 6 ⊢ (((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)) ↔ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
7	6	opabbii 5153	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))}
8	5, 7	eqtrdi 2788	. . . 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 5499	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))} ≠ ∅ ↔ ∃𝑥∃𝑦((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
11		exdistrv 1957	. . . 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 38875	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡(𝑅 ⋉ 𝑆)𝐵 ↔ ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅))
15		brcosscnv 38874	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
16		brcosscnv 38874	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑆𝐵 ↔ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
17	15, 16	anbi12d 633	. 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 1781   ∈ wcel 2114   ≠ wne 2933   ∩ cin 3889  ∅c0 4274   class class class wbr 5086  {copab 5148  ^◡ccnv 5621  [cec 8632   ⋉ cxrn 38486   ≀ ccoss 38495

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 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

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 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-1st 7933  df-2nd 7934  df-ec 8636  df-xrn 38692  df-coss 38813

This theorem is referenced by:  1cosscnvxrn  38877