Theorem cnvco 5835

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 1863	. . . 4 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
2		vex 3445	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3445	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	brco 5820	. . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		vex 3445	. . . . . . 7 ⊢ 𝑧 ∈ V
6	3, 5	brcnv 5832	. . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦)
7	5, 2	brcnv 5832	. . . . . 6 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧)
8	6, 7	anbi12i 629	. . . . 5 ⊢ ((𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ (𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
9	8	exbii 1850	. . . 4 ⊢ (∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
10	1, 4, 9	3bitr4i 303	. . 3 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥))
11	10	opabbii 5166	. 2 ⊢ {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
12		df-cnv 5633	. 2 ⊢ ◡(𝐴 ∘ 𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦}
13		df-co 5634	. 2 ⊢ (◡𝐵 ∘ ◡𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
14	11, 12, 13	3eqtr4i 2770	1 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   class class class wbr 5099  {copab 5161  ^◡ccnv 5624   ∘ ccom 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-cnv 5633  df-co 5634

This theorem is referenced by:  rncoss  5927  rncoeq  5932  dmco  6214  cores2  6219  co01  6221  coi2  6223  relcnvtrg  6226  dfdm2  6240  f1cof1  6741  cofunex2g  7897  fparlem3  8059  fparlem4  8060  suppco  8151  fsuppcolem  9309  relexpcnv  14963  relexpaddg  14981  cnvps  18506  gimco  19202  gsumzf1o  19846  cnco  23215  ptrescn  23588  qtopcn  23663  hmeoco  23721  cncombf  25620  deg1val  26062  fcoinver  32683  ofpreima  32747  cycpmconjv  33228  cycpmconjs  33242  cyc3conja  33243  esplysply  33740  mbfmco  34434  eulerpartlemmf  34545  cvmliftmolem1  35488  cvmlift2lem9a  35510  cvmlift2lem9  35518  mclsppslem  35790  ftc1anclem3  37909  trlcocnv  41059  tendoicl  41135  cdlemk45  41286  rimco  42851  cononrel1  43913  cononrel2  43914  cnvtrcl0  43945  cnvtrrel  43989  relexpaddss  44037  frege131d  44083  brco2f1o  44351  brco3f1o  44352  clsneicnv  44424  neicvgnvo  44434  smfco  47123  upgrimpthslem1  48230  upgrimspths  48233