Theorem ssbrd 5192

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 3982	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∈ 𝐴 → ⟨𝐶, 𝐷⟩ ∈ 𝐵))
3		df-br 5150	. 2 ⊢ (𝐶𝐴𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐴)
4		df-br 5150	. 2 ⊢ (𝐶𝐵𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝐵)
5	2, 3, 4	3imtr4g 296	1 ⊢ (𝜑 → (𝐶𝐴𝐷 → 𝐶𝐵𝐷))

Syntax hints:   → wi 4   ∈ wcel 2107   ⊆ wss 3949  ⟨cop 4635   class class class wbr 5149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-br 5150

This theorem is referenced by:  ssbr  5193  sess1  5645  brrelex12  5729  eqbrrdva  5870  predtrss  6324  ersym  8715  ertr  8718  ttrclss  9715  fpwwe2lem5  10630  fpwwe2lem6  10631  fpwwe2lem8  10633  fpwwe2lem11  10636  fpwwe2lem12  10637  fpwwe2  10638  coss12d  14919  fthres2  17883  invfuc  17927  pospo  18298  dirref  18554  efgcpbl  19624  frgpuplem  19640  subrguss  20334  znleval  21110  ustref  23723  ustuqtop4  23749  metider  32874  mclsppslem  34574  fundmpss  34738  eqvrelsym  37475  eqvreltr  37477  iunrelexpuztr  42470  frege96d  42500  frege91d  42502  frege98d  42504  frege124d  42512  grucollcld  43019