Theorem co01 6265

Description: Composition with the empty set. (Contributed by NM, 24-Apr-2004.)

Assertion

Ref	Expression
co01	⊢ (∅ ∘ 𝐴) = ∅

Proof of Theorem co01

Step	Hyp	Ref	Expression
1		cnv0 5871	. . . 4 ⊢ ◡∅ = ∅
2		cnvco 5877	. . . . 5 ⊢ ◡(∅ ∘ 𝐴) = (◡𝐴 ∘ ◡∅)
3	1	coeq2i 5848	. . . . 5 ⊢ (◡𝐴 ∘ ◡∅) = (◡𝐴 ∘ ∅)
4		co02 6264	. . . . 5 ⊢ (◡𝐴 ∘ ∅) = ∅
5	2, 3, 4	3eqtri 2792	. . . 4 ⊢ ◡(∅ ∘ 𝐴) = ∅
6	1, 5	eqtr4i 2791	. . 3 ⊢ ◡∅ = ◡(∅ ∘ 𝐴)
7	6	cnveqi 5862	. 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ 𝐴)
8		rel0 5787	. . 3 ⊢ Rel ∅
9		dfrel2 6189	. . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅)
10	8, 9	mpbi 233	. 2 ⊢ ◡◡∅ = ∅
11		relco 6112	. . 3 ⊢ Rel (∅ ∘ 𝐴)
12		dfrel2 6189	. . 3 ⊢ (Rel (∅ ∘ 𝐴) ↔ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴))
13	11, 12	mpbi 233	. 2 ⊢ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴)
14	7, 10, 13	3eqtr3ri 2797	1 ⊢ (∅ ∘ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570  ∅c0 4286  ^◡ccnv 5662   ∘ ccom 5667  Rel wrel 5668

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 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672

This theorem is used by:  xpcoid  6295  0trrel  15042  relexpsucrd  15094  relexpaddd  15115  gsumval3  20021  utop2nei  24458  cononrel2  44379  setc1ocofval  50329