Theorem cnvco1 36439

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 6095	. 2 ⊢ Rel ◡(◡𝐴 ∘ 𝐵)
2		relco 6099	. 2 ⊢ Rel (◡𝐵 ∘ 𝐴)
3		vex 3454	. . . . . . 7 ⊢ 𝑧 ∈ V
4		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
5	3, 4	brcnv 5857	. . . . . 6 ⊢ (𝑧◡𝐵𝑦 ↔ 𝑦𝐵𝑧)
6	5	bicomi 227	. . . . 5 ⊢ (𝑦𝐵𝑧 ↔ 𝑧◡𝐵𝑦)
7		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
8	3, 7	brcnv 5857	. . . . 5 ⊢ (𝑧◡𝐴𝑥 ↔ 𝑥𝐴𝑧)
9	6, 8	anbi12ci 641	. . . 4 ⊢ ((𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ (𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦))
10	9	exbii 1881	. . 3 ⊢ (∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦))
11	7, 4	opelcnv 5856	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑥⟩ ∈ (◡𝐴 ∘ 𝐵))
12	4, 7	opelco 5846	. . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ (◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥))
13	11, 12	bitri 278	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ∃𝑧(𝑦𝐵𝑧 ∧ 𝑧◡𝐴𝑥))
14	7, 4	opelco 5846	. . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝐵 ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡𝐵𝑦))
15	10, 13, 14	3bitr4i 306	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡(◡𝐴 ∘ 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (◡𝐵 ∘ 𝐴))
16	1, 2, 15	eqrelriiv 5763	1 ⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ⟨cop 4590   class class class wbr 5103  ^◡ccnv 5647   ∘ ccom 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657

This theorem is used by:  pprodcnveq  36561