Theorem coss1 5277

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 22	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
2	1	ssbrd 4696	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑧 → 𝑦𝐵𝑧))
3	2	anim2d 589	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
4	3	eximdv 1846	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) → ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
5	4	ssopab2dv 5004	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)})
6		df-co 5123	. 2 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)}
7		df-co 5123	. 2 ⊢ (𝐵 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}
8	5, 6, 7	3sstr4g 3646	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶))

Syntax hints:   → wi 4   ∧ wa 384  ∃wex 1704   ⊆ wss 3574   class class class wbr 4653  {copab 4712   ∘ ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-co 5123

This theorem is referenced by:  coeq1  5279  funss  5907  tposss  7353  rtrclreclem4  13801  tsrdir  17238  ustex2sym  22020  ustex3sym  22021  ustund  22025  ustneism  22027  trust  22033  utop2nei  22054  neipcfilu  22100  trclubgNEW  37925  trrelsuperrel2dg  37963  trclrelexplem  38003  cotrcltrcl  38017  cotrclrcl  38034  frege96d  38041  frege97d  38044  frege109d  38049  frege131d  38056