Theorem cnvco1 35981

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 6073	. 2 ⊢ Rel ◡(◡𝐴 ∘ 𝐵)
2		relco 6077	. 2 ⊢ Rel (◡𝐵 ∘ 𝐴)
3		vex 3446	. . . . . . 7 ⊢ 𝑧 ∈ V
4		vex 3446	. . . . . . 7 ⊢ 𝑦 ∈ V
5	3, 4	brcnv 5841	. . . . . 6 ⊢ (𝑧◡𝐵𝑦 ↔ 𝑦𝐵𝑧)
6	5	bicomi 224	. . . . 5 ⊢ (𝑦𝐵𝑧 ↔ 𝑧◡𝐵𝑦)
7		vex 3446	. . . . . 6 ⊢ 𝑥 ∈ V
8	3, 7	brcnv 5841	. . . . 5 ⊢ (𝑧◡𝐴𝑥 ↔ 𝑥𝐴𝑧)
9	6, 8	anbi12ci 630	. . . 4 ⊢ ((𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ (𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦))
10	9	exbii 1850	. . 3 ⊢ (∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦))
11	7, 4	opelcnv 5840	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐴 ∘ 𝐵))
12	4, 7	opelco 5830	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥))
13	11, 12	bitri 275	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥))
14	7, 4	opelco 5830	. . 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 5749	1 ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴)

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ⟨cop 4588   class class class wbr 5100  ^◡ccnv 5633   ∘ ccom 5638

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 5245  ax-pr 5381

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 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643

This theorem is referenced by:  pprodcnveq  36103