Theorem coss2 5721

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 5102	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦))
2	1	anim1d 612	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → (𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)))
3	2	eximdv 1914	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)))
4	3	ssopab2dv 5430	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)})
5		df-co 5558	. 2 ⊢ (𝐶 ∘ 𝐴) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)}
6		df-co 5558	. 2 ⊢ (𝐶 ∘ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)}
7	4, 5, 6	3sstr4g 4011	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵))

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1776   ⊆ wss 3935   class class class wbr 5058  {copab 5120   ∘ ccom 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-in 3942  df-ss 3951  df-br 5059  df-opab 5121  df-co 5558

This theorem is referenced by:  coeq2  5723  funss  6368  tposss  7887  dftpos4  7905  rtrclreclem4  14414  tsrdir  17842  mvdco  18567  ustex2sym  22819  ustex3sym  22820  ustneism  22826  trust  22832  utop2nei  22853  neipcfilu  22899  fcoinver  30351  trclubgNEW  39971  trrelsuperrel2dg  40009