Theorem brrelex12 5120

Description: A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brrelex12	⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem brrelex12

Step	Hyp	Ref	Expression
1		df-rel 5086	. . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 206	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	ssbrd 4661	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝐵 → 𝐴(V × V)𝐵))
4	3	imp 445	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴(V × V)𝐵)
5		brxp 5112	. 2 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
6	4, 5	sylib 208	1 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1992  Vcvv 3191   ⊆ wss 3560   class class class wbr 4618   × cxp 5077  Rel wrel 5084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086

This theorem is referenced by:  brrelex  5121  brrelex2  5122  nprrel12  5125  relbrcnvg  5467  ovprc  6637  oprabv  6657  brovex  7294  ersym  7700  relelec  7733  encv  7908  fsuppunbi  8241  fpwwe2lem2  9399  fpwwelem  9412  brfi1uzind  13214  brfi1uzindOLD  13220  isstruct2  15785  brssc  16390  cofuval2  16463  isfull  16486  isfth  16490  isnat  16523  pslem  17122  frgpuplem  18101  dvdsr  18562  ulmval  24033  perpln1  25500  perpln2  25501  opelco3  31372  rngoablo2  33326  aovprc  40559  aovrcl  40560