Theorem cnvco 5863

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 1883	. . . 4 ⊢ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
2		vex 3460	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3460	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	brco 5844	. . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		vex 3460	. . . . . . 7 ⊢ 𝑧 ∈ V
6	3, 5	brcnv 5856	. . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦)
7	5, 2	brcnv 5856	. . . . . 6 ⊢ (𝑧◡𝐵𝑥 ↔ 𝑥𝐵𝑧)
8	6, 7	anbi12i 637	. . . . 5 ⊢ ((𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ (𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
9	8	exbii 1870	. . . 4 ⊢ (∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥) ↔ ∃𝑧(𝑧𝐴𝑦 ∧ 𝑥𝐵𝑧))
10	1, 4, 9	3bitr4i 305	. . 3 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥))
11	10	opabbii 5169	. 2 ⊢ {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
12		df-cnv 5657	. 2 ⊢ ◡(𝐴 ∘ 𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦}
13		df-co 5658	. 2 ⊢ (◡𝐵 ∘ ◡𝐴) = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
14	11, 12, 13	3eqtr4i 2797	1 ⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)

Syntax hints:   ∧ wa 399   = wceq 1562  ∃wex 1801   class class class wbr 5102  {copab 5164  ^◡ccnv 5648   ∘ ccom 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-cnv 5657  df-co 5658

This theorem is referenced by:  rncoss  5955  rncoeq  5960  dmco  6244  cores2  6249  co01  6251  coi2  6253  relcnvtrg  6256  dfdm2  6270  f1cof1  6774  cofunex2g  7933  fparlem3  8095  fparlem4  8096  suppco  8188  fsuppcolem  9349  relexpcnv  15050  relexpaddg  15068  cnvps  18612  gimco  19310  gsumzf1o  19954  cnco  23328  ptrescn  23701  qtopcn  23776  hmeoco  23834  cncombf  25722  deg1val  26158  fcoinver  32806  ofpreima  32869  cycpmconjv  33324  cycpmconjs  33338  cyc3conja  33339  esplysply  33870  mbfmco  34563  eulerpartlemmf  34674  cvmliftmolem1  35636  cvmlift2lem9a  35658  cvmlift2lem9  35666  mclsppslem  35938  ftc1anclem3  38199  trlcocnv  41349  tendoicl  41425  cdlemk45  41576  rimco  43142  cononrel1  44175  cononrel2  44176  cnvtrcl0  44207  cnvtrrel  44251  relexpaddss  44299  frege131d  44345  brco2f1o  44613  brco3f1o  44614  clsneicnv  44686  neicvgnvo  44696  smfco  47381  upgrimpthslem1  48534  upgrimspths  48537