Theorem br1cosscnvxrn 39416

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 39258	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)})
2		ecxrn 39258	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → [𝐵](𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)})
3	1, 2	ineqan12d 4167	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)}))
4		inopab 5803	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))}
5	3, 4	eqtrdi 2811	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))})
6		an4 669	. . . . . 6 ⊢ (((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)) ↔ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
7	6	opabbii 5171	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))}
8	5, 7	eqtrdi 2811	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))})
9	8	neeq1d 3014	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))} ≠ ∅))
10		opabn0 5524	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))} ≠ ∅ ↔ ∃𝑥∃𝑦((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
11		exdistrv 1988	. . . 4 ⊢ (∃𝑥∃𝑦((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)) ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
12	10, 11	bitri 278	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))} ≠ ∅ ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
13	9, 12	bitrdi 290	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))))
14		brcosscnv2 39415	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡(𝑅 ⋉ 𝑆)𝐵 ↔ ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅))
15		brcosscnv 39414	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
16		brcosscnv 39414	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑆𝐵 ↔ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
17	15, 16	anbi12d 644	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ≀ ◡𝑅𝐵 ∧ 𝐴 ≀ ◡𝑆𝐵) ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))))
18	13, 14, 17	3bitr4d 314	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡(𝑅 ⋉ 𝑆)𝐵 ↔ (𝐴 ≀ ◡𝑅𝐵 ∧ 𝐴 ≀ ◡𝑆𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∃wex 1812   ∈ wcel 2145   ≠ wne 2955   ∩ cin 3897  ∅c0 4278   class class class wbr 5102  {copab 5166  ^◡ccnv 5646  [cec 8693   ⋉ cxrn 39026   ≀ ccoss 39035

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fo 6533  df-fv 6535  df-1st 7984  df-2nd 7985  df-ec 8697  df-xrn 39232  df-coss 39353

This theorem is used by:  1cosscnvxrn  39417