Theorem 1cosscnvxrn 38877

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 38876	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)))
2	1	el2v 3437	. . . 4 ⊢ (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦))
3	2	opabbii 5153	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)}
4		inopab 5776	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)}
5	3, 4	eqtr4i 2763	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦})
6		relcoss 38825	. . 3 ⊢ Rel ≀ ◡(𝐴 ⋉ 𝐵)
7		dfrel4v 6146	. . 3 ⊢ (Rel ≀ ◡(𝐴 ⋉ 𝐵) ↔ ≀ ◡(𝐴 ⋉ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦})
8	6, 7	mpbi 230	. 2 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦}
9		relcoss 38825	. . . 4 ⊢ Rel ≀ ◡𝐴
10		dfrel4v 6146	. . . 4 ⊢ (Rel ≀ ◡𝐴 ↔ ≀ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦})
11	9, 10	mpbi 230	. . 3 ⊢ ≀ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦}
12		relcoss 38825	. . . 4 ⊢ Rel ≀ ◡𝐵
13		dfrel4v 6146	. . . 4 ⊢ (Rel ≀ ◡𝐵 ↔ ≀ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦})
14	12, 13	mpbi 230	. . 3 ⊢ ≀ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦}
15	11, 14	ineq12i 4159	. 2 ⊢ ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦})
16	5, 8, 15	3eqtr4i 2770	1 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  Vcvv 3430   ∩ cin 3889   class class class wbr 5086  {copab 5148  ^◡ccnv 5621  Rel wrel 5627   ⋉ 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:  disjxrn  39158