Theorem cnvco 4789

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 1596	. . . 4 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
2		vex 2729	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 2729	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	brco 4775	. . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		vex 2729	. . . . . . 7 ⊢ 𝑧 ∈ V
6	3, 5	brcnv 4787	. . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦)
7	5, 2	brcnv 4787	. . . . . 6 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧)
8	6, 7	anbi12i 456	. . . . 5 ⊢ ((𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ (𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
9	8	exbii 1593	. . . 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 4049	. 2 ⊢ {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
12		df-cnv 4612	. 2 ⊢ ◡(𝐴 ∘ 𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦}
13		df-co 4613	. 2 ⊢ (◡𝐵 ∘ ◡𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
14	11, 12, 13	3eqtr4i 2196	1 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)

Syntax hints:   ∧ wa 103   = wceq 1343  ∃wex 1480   class class class wbr 3982  {copab 4042  ^◡ccnv 4603   ∘ ccom 4608

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-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  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-cnv 4612  df-co 4613

This theorem is referenced by:  rncoss  4874  rncoeq  4877  dmco  5112  cores2  5116  co01  5118  coi2  5120  relcnvtr  5123  dfdm2  5138  f1co  5405  cofunex2g  6078  caseinj  7054  djuinj  7071  cnco  12861  hmeoco  12956