Theorem ssbrd 3775

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 2917	. 2 ⊢ (φ → (⟨𝐶, 𝐷⟩ ∈ A → ⟨𝐶, 𝐷⟩ ∈ B))
3		df-br 3735	. 2 ⊢ (𝐶A𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ A)
4		df-br 3735	. 2 ⊢ (𝐶B𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ B)
5	2, 3, 4	3imtr4g 194	1 ⊢ (φ → (𝐶A𝐷 → 𝐶B𝐷))

Syntax hints:   → wi 4   ∈ wcel 1370   ⊆ wss 2890  ⟨cop 3349   class class class wbr 3734

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-11 1374  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-in 2897  df-ss 2904  df-br 3735

This theorem is referenced by:  ssbri  3776  sess1  4038  brrelex12  4304  coss1  4414  coss2  4415  eqbrrdva  4428  ersym  6025  ertr  6028