Theorem brel 5090

Description: Two things in a binary relation belong to the relation's domain. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypothesis

Ref	Expression
brel.1	⊢ 𝑅 ⊆ (𝐶 × 𝐷)

Assertion

Ref	Expression
brel	⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))

Proof of Theorem brel

Step	Hyp	Ref	Expression
1		brel.1	. . 3 ⊢ 𝑅 ⊆ (𝐶 × 𝐷)
2	1	ssbri 4627	. 2 ⊢ (𝐴𝑅𝐵 → 𝐴(𝐶 × 𝐷)𝐵)
3		brxp 5071	. 2 ⊢ (𝐴(𝐶 × 𝐷)𝐵 ↔ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))
4	2, 3	sylib 207	1 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷))

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ⊆ wss 3540   class class class wbr 4583   × cxp 5036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044

This theorem is referenced by:  brab2a  5091  brab2ga  5117  soirri  5441  sotri  5442  sotri2  5444  sotri3  5445  ndmovord  6722  ndmovordi  6723  swoer  7659  brecop2  7728  ecopovsym  7736  ecopovtrn  7737  hartogslem1  8330  nlt1pi  9607  indpi  9608  nqerf  9631  ordpipq  9643  lterpq  9671  ltexnq  9676  ltbtwnnq  9679  ltrnq  9680  prnmadd  9698  genpcd  9707  nqpr  9715  1idpr  9730  ltexprlem4  9740  ltexpri  9744  ltaprlem  9745  prlem936  9748  reclem2pr  9749  reclem3pr  9750  reclem4pr  9751  suplem1pr  9753  suplem2pr  9754  supexpr  9755  recexsrlem  9803  addgt0sr  9804  mulgt0sr  9805  mappsrpr  9808  map2psrpr  9810  supsrlem  9811  supsr  9812  ltresr  9840  dfle2  11856  dflt2  11857  dvdszrcl  14826  letsr  17050  hmphtop  21391  vcex  26817  brtxp2  31158  brpprod3a  31163