Theorem ssbrd 3971

Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)

Hypothesis

Ref	Expression
ssbrd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ssbrd	⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Proof of Theorem ssbrd

Step	Hyp	Ref	Expression
1		ssbrd.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2	1	sseld 3096	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∈ 𝐴 → ⟨𝐶, 𝐷⟩ ∈ 𝐵))
3		df-br 3930	. 2 ⊢ (𝐶𝐴𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐴)
4		df-br 3930	. 2 ⊢ (𝐶𝐵𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐵)
5	2, 3, 4	3imtr4g 204	1 ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Syntax hints:   → wi 4   ∈ wcel 1480   ⊆ wss 3071  ⟨cop 3530   class class class wbr 3929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084  df-br 3930

This theorem is referenced by:  ssbri  3972  sess1  4259  brrelex12  4577  coss1  4694  coss2  4695  eqbrrdva  4709  ersym  6441  ertr  6444