Theorem brelrng 3349

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 3322	. . . 4 |- ((B e. G /\ B`'CA) -> B e. dom `' C)
2	1	3adant1 799	. . 3 |- ((A e. F /\ B e. G /\ B`'CA) -> B e. dom `' C)
3		brcnvg 3303	. . . . . 6 |- ((B e. G /\ A e. F) -> (B`'CA <-> ACB))
4	3	ancoms 438	. . . . 5 |- ((A e. F /\ B e. G) -> (B`'CA <-> ACB))
5	4	biimprd 154	. . . 4 |- ((A e. F /\ B e. G) -> (ACB -> B`'CA))
6	5	3impia 832	. . 3 |- ((A e. F /\ B e. G /\ ACB) -> B`'CA)
7	2, 6	syld3an3 872	. 2 |- ((A e. F /\ B e. G /\ ACB) -> B e. dom `' C)
8		df-rn 3195	. 2 |- ran C = dom `' C
9	7, 8	syl6eleqr 1562	1 |- ((A e. F /\ B e. G /\ ACB) -> B e. ran C)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   e. wcel 960   class class class wbr 2624  `'ccnv 3175  dom cdm 3176  ran crn 3177

This theorem is referenced by:  brelrn 3350  relelrng 3353

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-cnv 3192  df-dm 3194  df-rn 3195

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