Theorem co01 2664

Description: Composition with the empty set.

Assertion

Ref	Expression
co01	⊢ (∅ ∘ A) = ∅

Proof of Theorem co01

Step	Hyp	Ref	Expression
1		cnv0 2633	. . . . . 6 ⊢ ◡∅ = ∅
2	1	coeq2i 2505	. . . . 5 ⊢ (◡A ∘ ◡∅) = (◡A ∘ ∅)
3		co02 2663	. . . . 5 ⊢ (◡A ∘ ∅) = ∅
4	2, 3	eqtr2 1120	. . . 4 ⊢ ∅ = (◡A ∘ ◡∅)
5		cnvco 2520	. . . 4 ⊢ ◡(∅ ∘ A) = (◡A ∘ ◡∅)
6	4, 1, 5	3eqtr4 1126	. . 3 ⊢ ◡∅ = ◡(∅ ∘ A)
7		cnveq 2513	. . 3 ⊢ (◡∅ = ◡(∅ ∘ A) → ◡◡∅ = ◡◡(∅ ∘ A))
8	6, 7	ax-mp 6	. 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ A)
9		rel0 2499	. . 3 ⊢ Rel ∅
10		dfrel2 2660	. . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅)
11	9, 10	mpbi 164	. 2 ⊢ ◡◡∅ = ∅
12		relco 2658	. . 3 ⊢ Rel (∅ ∘ A)
13		dfrel2 2660	. . 3 ⊢ (Rel (∅ ∘ A) ↔ ◡◡(∅ ∘ A) = (∅ ∘ A))
14	12, 13	mpbi 164	. 2 ⊢ ◡◡(∅ ∘ A) = (∅ ∘ A)
15	8, 11, 14	3eqtr3r 1125	1 ⊢ (∅ ∘ A) = ∅

Syntax hints:   = wceq 1091  ∅c0 1707  ^◡ccnv 2409   ∘ ccom 2414  Rel wrel 2415

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427

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