Theorem br1cosscnvxrn 34854

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 34779	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)})
2		ecxrn 34779	. . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → [𝐵](𝑅 ⋉ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)})
3	1, 2	ineqan12d 4039	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)}))
4		inopab 5498	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))}
5	3, 4	syl6eq 2830	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))})
6		an4 646	. . . . . 6 ⊢ (((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦)) ↔ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
7	6	opabbii 4953	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦) ∧ (𝐵𝑅𝑥 ∧ 𝐵𝑆𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))}
8	5, 7	syl6eq 2830	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) = {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))})
9	8	neeq1d 3028	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ {⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))} ≠ ∅))
10		opabn0 5243	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))} ≠ ∅ ↔ ∃𝑥∃𝑦((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
11		exdistrv 1998	. . . 4 ⊢ (∃𝑥∃𝑦((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)) ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
12	10, 11	bitri 267	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ (𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))} ≠ ∅ ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
13	9, 12	syl6bb 279	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅ ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))))
14		brcosscnv2 34853	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡(𝑅 ⋉ 𝑆)𝐵 ↔ ([𝐴](𝑅 ⋉ 𝑆) ∩ [𝐵](𝑅 ⋉ 𝑆)) ≠ ∅))
15		brcosscnv 34852	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑅𝐵 ↔ ∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥)))
16		brcosscnv 34852	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡𝑆𝐵 ↔ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦)))
17	15, 16	anbi12d 624	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ≀ ◡𝑅𝐵 ∧ 𝐴 ≀ ◡𝑆𝐵) ↔ (∃𝑥(𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥) ∧ ∃𝑦(𝐴𝑆𝑦 ∧ 𝐵𝑆𝑦))))
18	13, 14, 17	3bitr4d 303	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≀ ◡(𝑅 ⋉ 𝑆)𝐵 ↔ (𝐴 ≀ ◡𝑅𝐵 ∧ 𝐴 ≀ ◡𝑆𝐵)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∃wex 1823   ∈ wcel 2107   ≠ wne 2969   ∩ cin 3791  ∅c0 4141   class class class wbr 4886  {copab 4948  ^◡ccnv 5354  [cec 8024   ⋉ cxrn 34607   ≀ ccoss 34608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fo 6141  df-fv 6143  df-1st 7445  df-2nd 7446  df-ec 8028  df-xrn 34763  df-coss 34799

This theorem is referenced by:  1cosscnvxrn  34855