Theorem brelrng 3343

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 3316	. . . 4 |- ((B e. G /\ B`'CA) -> B e. dom `' C)
2	1	3adant1 797	. . 3 |- ((A e. F /\ B e. G /\ B`'CA) -> B e. dom `' C)
3		brcnvg 3297	. . . . . 6 |- ((B e. G /\ A e. F) -> (B`'CA <-> ACB))
4	3	ancoms 436	. . . . 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 830	. . 3 |- ((A e. F /\ B e. G /\ ACB) -> B`'CA)
7	2, 6	syld3an3 870	. 2 |- ((A e. F /\ B e. G /\ ACB) -> B e. dom `' C)
8		df-rn 3189	. 2 |- ran C = dom `' C
9	7, 8	syl6eleqr 1559	1 |- ((A e. F /\ B e. G /\ ACB) -> B e. ran C)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   e. wcel 958   class class class wbr 2619  `'ccnv 3169  dom cdm 3170  ran crn 3171

This theorem is referenced by:  brelrn 3344  relelrng 3347  spwpr3OLD 8662

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-cnv 3186  df-dm 3188  df-rn 3189

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