Theorem coss1 5842

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		ssbr 5159	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧))
2	1	anim2d 623	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
3	2	eximdv 1944	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
4	3	ssopab2dv 5537	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)})
5		df-co 5671	. 2 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)}
6		df-co 5671	. 2 ⊢ (𝐵 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}
7	4, 5, 6	3sstr4g 3998	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶))

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1806   ⊆ wss 3913   class class class wbr 5113  {copab 5177   ∘ ccom 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ss 3930  df-br 5114  df-opab 5178  df-co 5671

This theorem is referenced by:  coeq1  5844  funss  6556  tposss  8222  cottrcl  9687  frmin  9720  frrlem16  9729  rtrclreclem4  15097  tsrdir  18659  ustex2sym  24342  ustex3sym  24343  ustneism  24349  trust  24354  utop2nei  24375  neipcfilu  24420  trclubgNEW  44235  trrelsuperrel2dg  44288  trclrelexplem  44328  cotrcltrcl  44342  cotrclrcl  44359  frege96d  44366  frege97d  44369  frege109d  44374  frege131d  44381