Theorem cnvco 4764

Description: Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
cnvco	⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)

Proof of Theorem cnvco

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

Step	Hyp	Ref	Expression
1		exancom 1585	. . . 4 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
2		vex 2712	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 2712	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	brco 4750	. . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		vex 2712	. . . . . . 7 ⊢ 𝑧 ∈ V
6	3, 5	brcnv 4762	. . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦)
7	5, 2	brcnv 4762	. . . . . 6 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧)
8	6, 7	anbi12i 456	. . . . 5 ⊢ ((𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ (𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
9	8	exbii 1582	. . . 4 ⊢ (∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
10	1, 4, 9	3bitr4i 211	. . 3 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥))
11	10	opabbii 4027	. 2 ⊢ {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
12		df-cnv 4587	. 2 ⊢ ◡(𝐴 ∘ 𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦}
13		df-co 4588	. 2 ⊢ (◡𝐵 ∘ ◡𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
14	11, 12, 13	3eqtr4i 2185	1 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)

Syntax hints:   ∧ wa 103   = wceq 1332  ∃wex 1469   class class class wbr 3961  {copab 4020  ^◡ccnv 4578   ∘ ccom 4583

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-cnv 4587  df-co 4588

This theorem is referenced by:  rncoss  4849  rncoeq  4852  dmco  5087  cores2  5091  co01  5093  coi2  5095  relcnvtr  5098  dfdm2  5113  f1co  5380  cofunex2g  6050  caseinj  7019  djuinj  7036  cnco  12560  hmeoco  12655