Theorem coss2 4435

Description: Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)

Assertion

Ref	Expression
coss2	⊢ (A ⊆ B → (𝐶 ∘ A) ⊆ (𝐶 ∘ B))

Proof of Theorem coss2

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 ⊢ (A ⊆ B → A ⊆ B)
2	1	ssbrd 3796	. . . . 5 ⊢ (A ⊆ B → (xAy → xBy))
3	2	anim1d 319	. . . 4 ⊢ (A ⊆ B → ((xAy ∧ y𝐶z) → (xBy ∧ y𝐶z)))
4	3	eximdv 1757	. . 3 ⊢ (A ⊆ B → (∃y(xAy ∧ y𝐶z) → ∃y(xBy ∧ y𝐶z)))
5	4	ssopab2dv 4006	. 2 ⊢ (A ⊆ B → {⟨x, z⟩ ∣ ∃y(xAy ∧ y𝐶z)} ⊆ {⟨x, z⟩ ∣ ∃y(xBy ∧ y𝐶z)})
6		df-co 4297	. 2 ⊢ (𝐶 ∘ A) = {⟨x, z⟩ ∣ ∃y(xAy ∧ y𝐶z)}
7		df-co 4297	. 2 ⊢ (𝐶 ∘ B) = {⟨x, z⟩ ∣ ∃y(xBy ∧ y𝐶z)}
8	5, 6, 7	3sstr4g 2980	1 ⊢ (A ⊆ B → (𝐶 ∘ A) ⊆ (𝐶 ∘ B))

Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ⊆ wss 2911   class class class wbr 3755  {copab 3808   ∘ ccom 4292

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-co 4297

This theorem is referenced by:  coeq2  4437  funss  4863  tposss  5802  dftpos4  5819