Theorem brelrn 4191

Description: The second argument of a binary relation belongs to its range.

Hypotheses

Ref	Expression
brelrn.1	|- A e. _V
brelrn.2	|- B e. _V

Assertion

Ref	Expression
brelrn	|- (ACB -> B e. ran C)

Proof of Theorem brelrn

Step	Hyp	Ref	Expression
1		brelrn.2	. 2 |- B e. _V
2		brelrn.1	. . 3 |- A e. _V
3		brelrng 4190	. . 3 |- ((A e. _V /\ B e. _V /\ ACB) -> B e. ran C)
4	2, 3	mp3an1 1178	. 2 |- ((B e. _V /\ ACB) -> B e. ran C)
5	1, 4	mpan 759	1 |- (ACB -> B e. ran C)

Syntax hints:   -> wi 3   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  ran crn 3987

This theorem is referenced by:  opelrn 4192  dfco2a 4394  cores 4400  coresOLD 4401  dffun9 4450  funcnv 4475  cbvfo 4861  psdmrn 9991  ranfldrefc 14362  domrngref 14364  rnhmpha 14889  tailf 15633  tailmap 15636

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-cnv 4002  df-dm 4004  df-rn 4005

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