Theorem brelrng 4190

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

Assertion

Ref	Expression
brelrng	|- ((A e. F /\ B e. G /\ ACB) -> B e. ran C)

Proof of Theorem brelrng

Step	Hyp	Ref	Expression
1		breldmg 4162	. . . 4 |- ((B e. G /\ B`'CA) -> B e. dom `' C)
2	1	3adant1 894	. . 3 |- ((A e. F /\ B e. G /\ B`'CA) -> B e. dom `' C)
3		brcnvg 4142	. . . . 5 |- ((B e. G /\ A e. F) -> (B`'CA <-> ACB))
4	3	ancoms 484	. . . 4 |- ((A e. F /\ B e. G) -> (B`'CA <-> ACB))
5	4	biimp3ar 1195	. . 3 |- ((A e. F /\ B e. G /\ ACB) -> B`'CA)
6	2, 5	syld3an3 1142	. 2 |- ((A e. F /\ B e. G /\ ACB) -> B e. dom `' C)
7		df-rn 4005	. 2 |- ran C = dom `' C
8	6, 7	syl6eleqr 1982	1 |- ((A e. F /\ B e. G /\ ACB) -> B e. ran C)

Syntax hints:   -> wi 3   <-> wb 163   /\ w3a 858   e. wcel 1300   class class class wbr 3338  `'ccnv 3985  dom cdm 3986  ran crn 3987

This theorem is referenced by:  brelrn 4191  relelrng 4194  spwpr4c 10009  istail 15634

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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