Theorem 1cosscnvxrn 39300

Description: Cosets by the converse range Cartesian product. (Contributed by Peter Mazsa, 19-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)

Assertion

Ref	Expression
1cosscnvxrn	⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵)

Proof of Theorem 1cosscnvxrn

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

Step	Hyp	Ref	Expression
1		br1cosscnvxrn 39299	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)))
2	1	el2v 3460	. . . 4 ⊢ (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦))
3	2	opabbii 5176	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)}
4		inopab 5814	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)}
5	3, 4	eqtr4i 2788	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦})
6		relcoss 39248	. . 3 ⊢ Rel ≀ ◡(𝐴 ⋉ 𝐵)
7		dfrel4v 6187	. . 3 ⊢ (Rel ≀ ◡(𝐴 ⋉ 𝐵) ↔ ≀ ◡(𝐴 ⋉ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦})
8	6, 7	mpbi 233	. 2 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦}
9		relcoss 39248	. . . 4 ⊢ Rel ≀ ◡𝐴
10		dfrel4v 6187	. . . 4 ⊢ (Rel ≀ ◡𝐴 ↔ ≀ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦})
11	9, 10	mpbi 233	. . 3 ⊢ ≀ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦}
12		relcoss 39248	. . . 4 ⊢ Rel ≀ ◡𝐵
13		dfrel4v 6187	. . . 4 ⊢ (Rel ≀ ◡𝐵 ↔ ≀ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦})
14	12, 13	mpbi 233	. . 3 ⊢ ≀ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦}
15	11, 14	ineq12i 4167	. 2 ⊢ ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦})
16	5, 8, 15	3eqtr4i 2795	1 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  Vcvv 3453   ∩ cin 3901   class class class wbr 5107  {copab 5171  ^◡ccnv 5658  Rel wrel 5664   ⋉ cxrn 38909   ≀ ccoss 38918

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 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  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 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7989  df-2nd 7990  df-ec 8701  df-xrn 39115  df-coss 39236

This theorem is used by:  disjxrn  39581