Theorem cnvin 5073

Description: Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)

Assertion

Ref	Expression
cnvin	⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)

Proof of Theorem cnvin

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

Step	Hyp	Ref	Expression
1		df-cnv 4667	. . 3 ⊢ ◡(𝐴 ∩ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∩ 𝐵)𝑥}
2		inopab 4794	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)}
3		brin 4081	. . . . 5 ⊢ (𝑦(𝐴 ∩ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥))
4	3	opabbii 4096	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)}
5	2, 4	eqtr4i 2217	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∩ 𝐵)𝑥}
6	1, 5	eqtr4i 2217	. 2 ⊢ ◡(𝐴 ∩ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
7		df-cnv 4667	. . 3 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
8		df-cnv 4667	. . 3 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
9	7, 8	ineq12i 3358	. 2 ⊢ (◡𝐴 ∩ ◡𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
10	6, 9	eqtr4i 2217	1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)

Syntax hints:   ∧ wa 104   = wceq 1364   ∩ cin 3152   class class class wbr 4029  {copab 4089  ^◡ccnv 4658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667

This theorem is referenced by:  rnin  5075  dminxp  5110  imainrect  5111  cnvcnv  5118