Theorem coss1 4434

Description: Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)

Assertion

Ref	Expression
coss1	⊢ (A ⊆ B → (A ∘ 𝐶) ⊆ (B ∘ 𝐶))

Proof of Theorem coss1

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 → (yAz → yBz))
3	2	anim2d 320	. . . 4 ⊢ (A ⊆ B → ((x𝐶y ∧ yAz) → (x𝐶y ∧ yBz)))
4	3	eximdv 1757	. . 3 ⊢ (A ⊆ B → (∃y(x𝐶y ∧ yAz) → ∃y(x𝐶y ∧ yBz)))
5	4	ssopab2dv 4006	. 2 ⊢ (A ⊆ B → {⟨x, z⟩ ∣ ∃y(x𝐶y ∧ yAz)} ⊆ {⟨x, z⟩ ∣ ∃y(x𝐶y ∧ yBz)})
6		df-co 4297	. 2 ⊢ (A ∘ 𝐶) = {⟨x, z⟩ ∣ ∃y(x𝐶y ∧ yAz)}
7		df-co 4297	. 2 ⊢ (B ∘ 𝐶) = {⟨x, z⟩ ∣ ∃y(x𝐶y ∧ yBz)}
8	5, 6, 7	3sstr4g 2980	1 ⊢ (A ⊆ B → (A ∘ 𝐶) ⊆ (B ∘ 𝐶))

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-bnd 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:  coeq1  4436  funss  4863  tposss  5802