Theorem cnvco1 33722

Description: Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)

Assertion

Ref	Expression
cnvco1	⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴)

Proof of Theorem cnvco1

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 6011	. 2 ⊢ Rel ◡(◡𝐴 ∘ 𝐵)
2		relco 6147	. 2 ⊢ Rel (◡𝐵 ∘ 𝐴)
3		vex 3435	. . . . . . 7 ⊢ 𝑧 ∈ V
4		vex 3435	. . . . . . 7 ⊢ 𝑦 ∈ V
5	3, 4	brcnv 5790	. . . . . 6 ⊢ (𝑧◡𝐵𝑦 ↔ 𝑦𝐵𝑧)
6	5	bicomi 223	. . . . 5 ⊢ (𝑦𝐵𝑧 ↔ 𝑧◡𝐵𝑦)
7		vex 3435	. . . . . 6 ⊢ 𝑥 ∈ V
8	3, 7	brcnv 5790	. . . . 5 ⊢ (𝑧◡𝐴𝑥 ↔ 𝑥𝐴𝑧)
9	6, 8	anbi12ci 628	. . . 4 ⊢ ((𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ (𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦))
10	9	exbii 1854	. . 3 ⊢ (∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦))
11	7, 4	opelcnv 5789	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐴 ∘ 𝐵))
12	4, 7	opelco 5779	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥))
13	11, 12	bitri 274	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥))
14	7, 4	opelco 5779	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐵 ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦))
15	10, 13, 14	3bitr4i 303	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐵 ∘ 𝐴))
16	1, 2, 15	eqrelriiv 5699	1 ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴)

Syntax hints:   ∧ wa 396   = wceq 1542  ∃wex 1786   ∈ wcel 2110  ⟨cop 4573   class class class wbr 5079  ^◡ccnv 5589   ∘ ccom 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599

This theorem is referenced by:  pprodcnveq  34181