Theorem 1cosscnvxrn 35259

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 35258	. . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)))
2	1	el2v 3443	. . . 4 ⊢ (𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦 ↔ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦))
3	2	opabbii 5031	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)}
4		inopab 5590	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ≀ ◡𝐴𝑦 ∧ 𝑥 ≀ ◡𝐵𝑦)}
5	3, 4	eqtr4i 2821	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦} = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦})
6		relcoss 35212	. . 3 ⊢ Rel ≀ ◡(𝐴 ⋉ 𝐵)
7		dfrel4v 5926	. . 3 ⊢ (Rel ≀ ◡(𝐴 ⋉ 𝐵) ↔ ≀ ◡(𝐴 ⋉ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦})
8	6, 7	mpbi 231	. 2 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡(𝐴 ⋉ 𝐵)𝑦}
9		relcoss 35212	. . . 4 ⊢ Rel ≀ ◡𝐴
10		dfrel4v 5926	. . . 4 ⊢ (Rel ≀ ◡𝐴 ↔ ≀ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦})
11	9, 10	mpbi 231	. . 3 ⊢ ≀ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦}
12		relcoss 35212	. . . 4 ⊢ Rel ≀ ◡𝐵
13		dfrel4v 5926	. . . 4 ⊢ (Rel ≀ ◡𝐵 ↔ ≀ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦})
14	12, 13	mpbi 231	. . 3 ⊢ ≀ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦}
15	11, 14	ineq12i 4109	. 2 ⊢ ( ≀ ◡𝐴 ∩ ≀ ◡𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐴𝑦} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ≀ ◡𝐵𝑦})
16	5, 8, 15	3eqtr4i 2828	1 ⊢ ≀ ◡(𝐴 ⋉ 𝐵) = ( ≀ ◡𝐴 ∩ ≀ ◡𝐵)

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1522  Vcvv 3436   ∩ cin 3860   class class class wbr 4964  {copab 5026  ^◡ccnv 5445  Rel wrel 5451   ⋉ cxrn 34997   ≀ ccoss 34998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fo 6234  df-fv 6236  df-1st 7548  df-2nd 7549  df-ec 8144  df-xrn 35167  df-coss 35203

This theorem is referenced by:  disjxrn  35521