Theorem co01 5184

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 5073	. . . 4 ⊢ ◡∅ = ∅
2		cnvco 4851	. . . . 5 ⊢ ◡(∅ ∘ 𝐴) = (◡𝐴 ∘ ◡∅)
3	1	coeq2i 4826	. . . . 5 ⊢ (◡𝐴 ∘ ◡∅) = (◡𝐴 ∘ ∅)
4		co02 5183	. . . . 5 ⊢ (◡𝐴 ∘ ∅) = ∅
5	2, 3, 4	3eqtri 2221	. . . 4 ⊢ ◡(∅ ∘ 𝐴) = ∅
6	1, 5	eqtr4i 2220	. . 3 ⊢ ◡∅ = ◡(∅ ∘ 𝐴)
7	6	cnveqi 4841	. 2 ⊢ ◡◡∅ = ◡◡(∅ ∘ 𝐴)
8		rel0 4788	. . 3 ⊢ Rel ∅
9		dfrel2 5120	. . 3 ⊢ (Rel ∅ ↔ ◡◡∅ = ∅)
10	8, 9	mpbi 145	. 2 ⊢ ◡◡∅ = ∅
11		relco 5168	. . 3 ⊢ Rel (∅ ∘ 𝐴)
12		dfrel2 5120	. . 3 ⊢ (Rel (∅ ∘ 𝐴) ↔ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴))
13	11, 12	mpbi 145	. 2 ⊢ ◡◡(∅ ∘ 𝐴) = (∅ ∘ 𝐴)
14	7, 10, 13	3eqtr3ri 2226	1 ⊢ (∅ ∘ 𝐴) = ∅

Syntax hints:   = wceq 1364  ∅c0 3450  ^◡ccnv 4662   ∘ ccom 4667  Rel wrel 4668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672

This theorem is referenced by: (None)