Theorem cnvco2 35722

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

Assertion

Ref	Expression
cnvco2	⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴)

Proof of Theorem cnvco2

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

Step	Hyp	Ref	Expression
1		relcnv 6134	. 2 ⊢ Rel ◡(𝐴 ∘ ◡𝐵)
2		relco 6138	. 2 ⊢ Rel (𝐵 ∘ ◡𝐴)
3		vex 3492	. . . . . 6 ⊢ 𝑦 ∈ V
4		vex 3492	. . . . . 6 ⊢ 𝑧 ∈ V
5	3, 4	brcnv 5907	. . . . 5 ⊢ (𝑦◡𝐵𝑧 ↔ 𝑧𝐵𝑦)
6		vex 3492	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5907	. . . . . 6 ⊢ (𝑥◡𝐴𝑧 ↔ 𝑧𝐴𝑥)
8	7	bicomi 224	. . . . 5 ⊢ (𝑧𝐴𝑥 ↔ 𝑥◡𝐴𝑧)
9	5, 8	anbi12ci 628	. . . 4 ⊢ ((𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ (𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦))
10	9	exbii 1846	. . 3 ⊢ (∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦))
11	6, 3	opelcnv 5906	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∘ ◡𝐵))
12	3, 6	opelco 5896	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∘ ◡𝐵) ↔ ∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥))
13	11, 12	bitri 275	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥))
14	6, 3	opelco 5896	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ∘ ◡𝐴) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦))
15	10, 13, 14	3bitr4i 303	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵 ∘ ◡𝐴))
16	1, 2, 15	eqrelriiv 5814	1 ⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴)

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ⟨cop 4654   class class class wbr 5166  ^◡ccnv 5699   ∘ ccom 5704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709

This theorem is referenced by: (None)