Theorem coss1 4873

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 4681	. . . . 5 ⊢ (A ⊆ B → (yAz → yBz))
3	2	anim2d 548	. . . 4 ⊢ (A ⊆ B → ((xCy ∧ yAz) → (xCy ∧ yBz)))
4	3	eximdv 1622	. . 3 ⊢ (A ⊆ B → (∃y(xCy ∧ yAz) → ∃y(xCy ∧ yBz)))
5	4	ssopab2dv 4716	. 2 ⊢ (A ⊆ B → {⟨x, z⟩ ∣ ∃y(xCy ∧ yAz)} ⊆ {⟨x, z⟩ ∣ ∃y(xCy ∧ yBz)})
6		df-co 4727	. 2 ⊢ (A ∘ C) = {⟨x, z⟩ ∣ ∃y(xCy ∧ yAz)}
7		df-co 4727	. 2 ⊢ (B ∘ C) = {⟨x, z⟩ ∣ ∃y(xCy ∧ yBz)}
8	5, 6, 7	3sstr4g 3313	1 ⊢ (A ⊆ B → (A ∘ C) ⊆ (B ∘ C))

Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   ⊆ wss 3258  {copab 4623   class class class wbr 4640   ∘ ccom 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-co 4727

This theorem is referenced by:  coeq1  4875  funss  5127