Theorem coss1 5843

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 5157	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧))
2	1	anim2d 624	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
3	2	eximdv 1950	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
4	3	ssopab2dv 5538	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)})
5		df-co 5672	. 2 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)}
6		df-co 5672	. 2 ⊢ (𝐵 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}
7	4, 5, 6	3sstr4g 3991	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃wex 1812   ⊆ wss 3906   class class class wbr 5111  {copab 5175   ∘ ccom 5667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ss 3923  df-br 5112  df-opab 5176  df-co 5672

This theorem is used by:  coeq1  5845  funss  6559  tposss  8229  cottrcl  9695  frmin  9728  frrlem16  9737  rtrclreclem4  15122  tsrdir  18682  ustex2sym  24425  ustex3sym  24426  ustneism  24432  trust  24437  utop2nei  24458  neipcfilu  24503  trclubgNEW  44402  trrelsuperrel2dg  44455  trclrelexplem  44495  cotrcltrcl  44509  cotrclrcl  44526  frege96d  44533  frege97d  44536  frege109d  44541  frege131d  44548