Theorem ghmghmrn 13930

Description: A group homomorphism from G to H is also a group homomorphism from G to its image in H. (Contributed by Paul Chapman, 3-Mar-2008.) (Revised by AV, 26-Aug-2021.)

Hypothesis

Ref	Expression
ghmghmrn.u	|- U = ( T ↾s ran F )

Assertion

Ref	Expression
ghmghmrn	|- ( F e. ( S GrpHom T ) -> F e. ( S GrpHom U ) )

Proof of Theorem ghmghmrn

Step	Hyp	Ref	Expression
1		ghmrn 13924	. 2 |- ( F e. ( S GrpHom T ) -> ran F e. (SubGrp ` T ) )
2		ssid 3248	. . . 4 |- ran F C_ ran F
3		ghmghmrn.u	. . . . 5 |- U = ( T ↾s ran F )
4	3	resghm2b 13929	. . . 4 |- ( ( ran F e. (SubGrp ` T ) /\ ran F C_ ran F ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) )
5	2, 4	mpan2 425	. . 3 |- ( ran F e. (SubGrp ` T ) -> ( F e. ( S GrpHom T ) <-> F e. ( S GrpHom U ) ) )
6	5	biimpd 144	. 2 |- ( ran F e. (SubGrp ` T ) -> ( F e. ( S GrpHom T ) -> F e. ( S GrpHom U ) ) )
7	1, 6	mpcom 36	1 |- ( F e. ( S GrpHom T ) -> F e. ( S GrpHom U ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2202   C_ wss 3201   ran crn 4732   ` cfv 5333  (class class class)co 6028   ↾_s cress 13163  SubGrpcsubg 13834   GrpHom cghm 13907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-mhm 13622  df-submnd 13623  df-grp 13666  df-minusg 13667  df-subg 13837  df-ghm 13908

This theorem is referenced by: (None)