Theorem cnvco 4940

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 1657	. . . 4 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
2		vex 2816	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 2816	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	brco 4926	. . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		vex 2816	. . . . . . 7 ⊢ 𝑧 ∈ V
6	3, 5	brcnv 4938	. . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦)
7	5, 2	brcnv 4938	. . . . . 6 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧)
8	6, 7	anbi12i 460	. . . . 5 ⊢ ((𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ (𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
9	8	exbii 1654	. . . 4 ⊢ (∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
10	1, 4, 9	3bitr4i 212	. . 3 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥))
11	10	opabbii 4177	. 2 ⊢ {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
12		df-cnv 4757	. 2 ⊢ ◡(𝐴 ∘ 𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦}
13		df-co 4758	. 2 ⊢ (◡𝐵 ∘ ◡𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
14	11, 12, 13	3eqtr4i 2263	1 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   class class class wbr 4109  {copab 4170  ^◡ccnv 4748   ∘ ccom 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-cnv 4757  df-co 4758

This theorem is referenced by:  rncoss  5028  rncoeq  5031  dmco  5271  cores2  5275  co01  5277  coi2  5279  relcnvtr  5282  dfdm2  5297  f1co  5585  cofunex2g  6303  suppcofn  6466  caseinj  7380  djuinj  7397  cnco  15086  hmeoco  15181