Theorem ghomcl 10392

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 10391	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:X-onto->Z)
6		fof 3672	. 2 |- (F:X-onto->Z -> F:X-->Z)
7		ffvelrn 3814	. . 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 775   = wceq 956   e. wcel 958   X. cxp 3168  ran crn 3171   |` cres 3172  -->wf 3178  -onto->wfo 3180  ` cfv 3182  (class class class)co 3963  Grpcgr 8033   GrpHom cghom 10378

This theorem is referenced by:  ghomgsg 10395

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-9 965  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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

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-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-opr 3965  df-oprab 3966  df-grp 8037  df-gid 8038  df-ginv 8039  df-subg 8115  df-ghom 10380

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