Theorem coss2 5813

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		ssbr 5144	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦))
2	1	anim1d 612	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → (𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)))
3	2	eximdv 1919	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)))
4	3	ssopab2dv 5507	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)})
5		df-co 5641	. 2 ⊢ (𝐶 ∘ 𝐴) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)}
6		df-co 5641	. 2 ⊢ (𝐶 ∘ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)}
7	4, 5, 6	3sstr4g 3989	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1781   ⊆ wss 3903   class class class wbr 5100  {copab 5162   ∘ ccom 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ss 3920  df-br 5101  df-opab 5163  df-co 5641

This theorem is referenced by:  coeq2  5815  funss  6519  tposss  8179  dftpos4  8197  ttrclco  9639  frmin  9673  frrlem16  9682  rtrclreclem4  14996  tsrdir  18539  mvdco  19386  ustex2sym  24173  ustex3sym  24174  ustneism  24180  trust  24185  utop2nei  24206  neipcfilu  24251  fcoinver  32690  trclubgNEW  43971  trrelsuperrel2dg  44024