Theorem cnvin 5011

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 4612	. . 3 ⊢ ◡(𝐴 ∩ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∩ 𝐵)𝑥}
2		inopab 4736	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)}
3		brin 4034	. . . . 5 ⊢ (𝑦(𝐴 ∩ 𝐵)𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥))
4	3	opabbii 4049	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∩ 𝐵)𝑥} = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥 ∧ 𝑦𝐵𝑥)}
5	2, 4	eqtr4i 2189	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴 ∩ 𝐵)𝑥}
6	1, 5	eqtr4i 2189	. 2 ⊢ ◡(𝐴 ∩ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
7		df-cnv 4612	. . 3 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
8		df-cnv 4612	. . 3 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
9	7, 8	ineq12i 3321	. 2 ⊢ (◡𝐴 ∩ ◡𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
10	6, 9	eqtr4i 2189	1 ⊢ ◡(𝐴 ∩ 𝐵) = (◡𝐴 ∩ ◡𝐵)

Syntax hints:   ∧ wa 103   = wceq 1343   ∩ cin 3115   class class class wbr 3982  {copab 4042  ^◡ccnv 4603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612

This theorem is referenced by:  rnin  5013  dminxp  5048  imainrect  5049  cnvcnv  5056