Theorem coss2 4877

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		id 19	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
2	1	ssbrd 4125	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦))
3	2	anim1d 336	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → (𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)))
4	3	eximdv 1926	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧) → ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)))
5	4	ssopab2dv 4366	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)} ⊆ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)})
6		df-co 4727	. 2 ⊢ (𝐶 ∘ 𝐴) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐴𝑦 ∧ 𝑦𝐶𝑧)}
7		df-co 4727	. 2 ⊢ (𝐶 ∘ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐵𝑦 ∧ 𝑦𝐶𝑧)}
8	5, 6, 7	3sstr4g 3267	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104  ∃wex 1538   ⊆ wss 3197   class class class wbr 4082  {copab 4143   ∘ ccom 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3203  df-ss 3210  df-br 4083  df-opab 4145  df-co 4727

This theorem is referenced by:  coeq2  4879  funss  5336  tposss  6390  dftpos4  6407