Theorem cnvco2 32996

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 5967	. 2 ⊢ Rel ◡(𝐴 ∘ ◡𝐵)
2		relco 6097	. 2 ⊢ Rel (𝐵 ∘ ◡𝐴)
3		vex 3497	. . . . . 6 ⊢ 𝑦 ∈ V
4		vex 3497	. . . . . 6 ⊢ 𝑧 ∈ V
5	3, 4	brcnv 5753	. . . . 5 ⊢ (𝑦◡𝐵𝑧 ↔ 𝑧𝐵𝑦)
6		vex 3497	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6, 4	brcnv 5753	. . . . . 6 ⊢ (𝑥◡𝐴𝑧 ↔ 𝑧𝐴𝑥)
8	7	bicomi 226	. . . . 5 ⊢ (𝑧𝐴𝑥 ↔ 𝑥◡𝐴𝑧)
9	5, 8	anbi12ci 629	. . . 4 ⊢ ((𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ (𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦))
10	9	exbii 1848	. . 3 ⊢ (∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦))
11	6, 3	opelcnv 5752	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 ∘ ◡𝐵))
12	3, 6	opelco 5742	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (𝐴 ∘ ◡𝐵) ↔ ∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥))
13	11, 12	bitri 277	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ∃𝑧(𝑦◡𝐵𝑧 ∧ 𝑧𝐴𝑥))
14	6, 3	opelco 5742	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 ∘ ◡𝐴) ↔ ∃𝑧(𝑥◡𝐴𝑧 ∧ 𝑧𝐵𝑦))
15	10, 13, 14	3bitr4i 305	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(𝐴 ∘ ◡𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐵 ∘ ◡𝐴))
16	1, 2, 15	eqrelriiv 5663	1 ⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴)

Syntax hints:   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5066  ^◡ccnv 5554   ∘ ccom 5559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564

This theorem is referenced by: (None)