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Theorem cnvin 4826
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 (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem cnvin
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 4436 . . 3 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴𝐵)𝑥}
2 inopab 4556 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥𝑦𝐵𝑥)}
3 brin 3884 . . . . 5 (𝑦(𝐴𝐵)𝑥 ↔ (𝑦𝐴𝑥𝑦𝐵𝑥))
43opabbii 3897 . . . 4 {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴𝐵)𝑥} = {⟨𝑥, 𝑦⟩ ∣ (𝑦𝐴𝑥𝑦𝐵𝑥)}
52, 4eqtr4i 2111 . . 3 ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) = {⟨𝑥, 𝑦⟩ ∣ 𝑦(𝐴𝐵)𝑥}
61, 5eqtr4i 2111 . 2 (𝐴𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
7 df-cnv 4436 . . 3 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
8 df-cnv 4436 . . 3 𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
97, 8ineq12i 3197 . 2 (𝐴𝐵) = ({⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
106, 9eqtr4i 2111 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  cin 2996   class class class wbr 3837  {copab 3890  ccnv 4427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436
This theorem is referenced by:  rnin  4828  dminxp  4862  imainrect  4863  cnvcnv  4870
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