Theorem ghomgrpi 10382

Description: The image of a group homomorphism from G to H is a subgroup of H (inference version). (Contributed by Paul Chapman, 25-Feb-2008.)

Hypotheses

Ref	Expression
ghomgrpi.1	|- G e. Grp
ghomgrpi.2	|- H e. Grp
ghomgrpi.3	|- F e. (G GrpHom H)
ghomgrpi.4	|- Y = ran F
ghomgrpi.5	|- S = (H |` (Y X. Y))

Assertion

Ref	Expression
ghomgrpi	|- S e. (SubGrp` H)

Proof of Theorem ghomgrpi

Step	Hyp	Ref	Expression
1		ghomgrpi.1	. 2 |- G e. Grp
2		ghomgrpi.2	. 2 |- H e. Grp
3		ghomgrpi.3	. 2 |- F e. (G GrpHom H)
4		eqid 1478	. 2 |- ran G = ran G
5		eqid 1478	. 2 |- (Id` G) = (Id` G)
6		eqid 1478	. 2 |- (inv` G) = (inv` G)
7		eqid 1478	. 2 |- ran H = ran H
8		eqid 1478	. 2 |- (Id` H) = (Id` H)
9		eqid 1478	. 2 |- (inv` H) = (inv` H)
10		ghomgrpi.4	. 2 |- Y = ran F
11		ghomgrpi.5	. 2 |- S = (H |` (Y X. Y))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	ghomgrpilem2 10381	1 |- S e. (SubGrp` H)

Syntax hints:   = wceq 958   e. wcel 960   X. cxp 3174  ran crn 3177   |` cres 3178  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031  invcgn 8032  SubGrpcsubg 8110   GrpHom cghom 10373

This theorem is referenced by:  ghomgrplem 10384  cayleylem3 10406

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-9 967  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-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

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-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  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-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-oprab 3972  df-grp 8034  df-gid 8035  df-ginv 8036  df-subg 8111  df-ghom 10375

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