Theorem cnvco 5838

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 3434	. . . . 5 ⊢ 𝑥 ∈ V
3		vex 3434	. . . . 5 ⊢ 𝑦 ∈ V
4	2, 3	brco 5823	. . . 4 ⊢ (𝑥(𝐴 ∘ 𝐵)𝑦 ↔ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦))
5		vex 3434	. . . . . . 7 ⊢ 𝑧 ∈ V
6	3, 5	brcnv 5835	. . . . . 6 ⊢ (𝑦◡𝐴𝑧 ↔ 𝑧𝐴𝑦)
7	5, 2	brcnv 5835	. . . . . 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 5153	. 2 ⊢ {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦} = {⟨𝑦, 𝑥⟩ ∣ ∃𝑧(𝑦◡𝐴𝑧 ∧ 𝑧◡𝐵𝑥)}
12		df-cnv 5636	. 2 ⊢ ◡(𝐴 ∘ 𝐵) = {⟨𝑦, 𝑥⟩ ∣ 𝑥(𝐴 ∘ 𝐵)𝑦}
13		df-co 5637	. 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 5086  {copab 5148  ^◡ccnv 5627   ∘ ccom 5632

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 5232  ax-pr 5374

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5636  df-co 5637

This theorem is referenced by:  rncoss  5930  rncoeq  5935  dmco  6217  cores2  6222  co01  6224  coi2  6226  relcnvtrg  6229  dfdm2  6243  f1cof1  6744  cofunex2g  7900  fparlem3  8061  fparlem4  8062  suppco  8153  fsuppcolem  9311  relexpcnv  14994  relexpaddg  15012  cnvps  18541  gimco  19240  gsumzf1o  19884  cnco  23228  ptrescn  23601  qtopcn  23676  hmeoco  23734  cncombf  25622  deg1val  26058  fcoinver  32671  ofpreima  32735  cycpmconjv  33200  cycpmconjs  33214  cyc3conja  33215  esplysply  33712  mbfmco  34405  eulerpartlemmf  34516  cvmliftmolem1  35460  cvmlift2lem9a  35482  cvmlift2lem9  35490  mclsppslem  35762  ftc1anclem3  38013  trlcocnv  41163  tendoicl  41239  cdlemk45  41390  rimco  42960  cononrel1  44018  cononrel2  44019  cnvtrcl0  44050  cnvtrrel  44094  relexpaddss  44142  frege131d  44188  brco2f1o  44456  brco3f1o  44457  clsneicnv  44529  neicvgnvo  44539  smfco  47227  upgrimpthslem1  48374  upgrimspths  48377