Theorem ghomcl 10387

Description: Closure of a group homomorphism. (Contributed by Paul Chapman, 3-Mar-2008.)

Hypotheses

Ref	Expression
ghomfo.1	|- X = ran G
ghomfo.2	|- Y = ran F
ghomfo.3	|- S = (H |` (Y X. Y))
ghomfo.4	|- Z = ran S

Assertion

Ref	Expression
ghomcl	|- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (A e. X -> (F` A) e. Z))

Proof of Theorem ghomcl

Step	Hyp	Ref	Expression
1		ghomfo.1	. . 3 |- X = ran G
2		ghomfo.2	. . 3 |- Y = ran F
3		ghomfo.3	. . 3 |- S = (H |` (Y X. Y))
4		ghomfo.4	. . 3 |- Z = ran S
5	1, 2, 3, 4	ghomfo 10386	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-onto->Z)
6		fof 3678	. 2 |- (F:X-onto->Z -> F:X-->Z)
7		ffvelrn 3820	. . 3 |- ((F:X-->Z /\ A e. X) -> (F` A) e. Z)
8	7	ex 373	. 2 |- (F:X-->Z -> (A e. X -> (F` A) e. Z))
9	5, 6, 8	3syl 20	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (A e. X -> (F` A) e. Z))

Syntax hints:   -> wi 3   /\ w3a 777   = wceq 958   e. wcel 960   X. cxp 3174  ran crn 3177   |` cres 3178  -->wf 3184  -onto->wfo 3186  ` cfv 3188  (class class class)co 3969  Grpcgr 8030   GrpHom cghom 10373

This theorem is referenced by:  ghomgsg 10390

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-nul 2715  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-if 2366  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-f1 3201  df-fo 3202  df-f1o 3203  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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