Theorem ghomid 10389

Description: A group homomorphism maps identity element to identity element. (Contributed by Paul Chapman, 3-Mar-2008.)

Hypotheses

Ref	Expression
ghomid.1	|- U = (Id` G)
ghomid.2	|- T = (Id` H)

Assertion

Ref	Expression
ghomid	|- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F` U) = T)

Proof of Theorem ghomid

Step	Hyp	Ref	Expression
1		eqid 1478	. . . . . . 7 |- ran G = ran G
2		ghomid.1	. . . . . . 7 |- U = (Id` G)
3	1, 2	grpidcl 8055	. . . . . 6 |- (G e. Grp -> U e. ran G)
4	3	3ad2ant1 802	. . . . 5 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> U e. ran G)
5	4, 4	jca 288	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (U e. ran G /\ U e. ran G))
6	1	ghomlin 10388	. . . 4 |- (((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) /\ (U e. ran G /\ U e. ran G)) -> ((F` U)H(F` U)) = (F` (UGU)))
7	5, 6	mpdan 706	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ((F` U)H(F` U)) = (F` (UGU)))
8	1, 2	grplid 8057	. . . . . 6 |- ((G e. Grp /\ U e. ran G) -> (UGU) = U)
9	3, 8	mpdan 706	. . . . 5 |- (G e. Grp -> (UGU) = U)
10	9	fveq2d 3734	. . . 4 |- (G e. Grp -> (F` (UGU)) = (F` U))
11	10	3ad2ant1 802	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F` (UGU)) = (F` U))
12	7, 11	eqtrd 1510	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ((F` U)H(F` U)) = (F` U))
13		ffvelrn 3820	. . . 4 |- ((F:ran G-->ran H /\ U e. ran G) -> (F` U) e. ran H)
14		eqid 1478	. . . . . . 7 |- ran H = ran H
15	1, 14	elghom 10379	. . . . . 6 |- ((G e. Grp /\ H e. Grp) -> (F e. (G GrpHom H) <-> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy)))))
16	15	biimp3a 921	. . . . 5 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F:ran G-->ran H /\ A.x e. ran GA.y e. ran G((F` x)H(F` y)) = (F` (xGy))))
17	16	pm3.26d 321	. . . 4 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> F:ran G-->ran H)
18	13, 17, 4	sylanc 473	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F` U) e. ran H)
19		ghomid.2	. . . . . 6 |- T = (Id` H)
20	14, 19	grpid 8061	. . . . 5 |- ((H e. Grp /\ (F` U) e. ran H) -> ((F` U) = T <-> ((F` U)H(F` U)) = (F` U)))
21	20	ex 373	. . . 4 |- (H e. Grp -> ((F` U) e. ran H -> ((F` U) = T <-> ((F` U)H(F` U)) = (F` U))))
22	21	3ad2ant2 803	. . 3 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ((F` U) e. ran H -> ((F` U) = T <-> ((F` U)H(F` U)) = (F` U))))
23	18, 22	mpd 26	. 2 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> ((F` U) = T <-> ((F` U)H(F` U)) = (F` U)))
24	12, 23	mpbird 196	1 |- ((G e. Grp /\ H e. Grp /\ F e. (G GrpHom H)) -> (F` U) = T)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  ran crn 3177  -->wf 3184  ` cfv 3188  (class class class)co 3969  Grpcgr 8030  Idcgi 8031   GrpHom cghom 10373

This theorem is referenced by:  ghomf1olem 10391

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-ghom 10375

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