Theorem 1cosscnvxrn 37333

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 37332	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)))
2	1	el2v 3482	. . . 4 ⊢ (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦))
3	2	opabbii 5214	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)}
4		inopab 5827	. . 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 37281	. . 3 ⊢ Rel ≀ ◡(𝐴 ⋉ 𝐵)
7		dfrel4v 6186	. . 3 ⊢ (Rel ≀ ◡(𝐴 ⋉ 𝐵) ↔ ≀ ◡(𝐴 ⋉ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦})
8	6, 7	mpbi 229	. 2 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦}
9		relcoss 37281	. . . 4 ⊢ Rel ≀ ◡𝐴
10		dfrel4v 6186	. . . 4 ⊢ (Rel ≀ ◡𝐴 ↔ ≀ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦})
11	9, 10	mpbi 229	. . 3 ⊢ ≀ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦}
12		relcoss 37281	. . . 4 ⊢ Rel ≀ ◡𝐵
13		dfrel4v 6186	. . . 4 ⊢ (Rel ≀ ◡𝐵 ↔ ≀ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦})
14	12, 13	mpbi 229	. . 3 ⊢ ≀ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦}
15	11, 14	ineq12i 4209	. 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 205   ∧ wa 396   = wceq 1541  Vcvv 3474   ∩ cin 3946   class class class wbr 5147  {copab 5209  ^◡ccnv 5674  Rel wrel 5680   ⋉ cxrn 37030   ≀ ccoss 37031

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7971  df-2nd 7972  df-ec 8701  df-xrn 37229  df-coss 37269

This theorem is referenced by:  disjxrn  37604