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Theorem ghomgsg 25104
 Description: A group homomorphism from to is also a group homomorphism from to its image in . (Contributed by Paul Chapman, 3-Mar-2008.)
Hypotheses
Ref Expression
ghomgsg.1
ghomgsg.2
Assertion
Ref Expression
ghomgsg GrpOpHom GrpOpHom

Proof of Theorem ghomgsg
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . . 4
2 ghomgsg.1 . . . 4
3 ghomgsg.2 . . . 4
4 eqid 2436 . . . 4
51, 2, 3, 4ghomfo 25102 . . 3 GrpOpHom
6 fof 5653 . . 3
75, 6syl 16 . 2 GrpOpHom
8 eqid 2436 . . . . . 6
91, 8elghom 21951 . . . . 5 GrpOpHom
109biimp3a 1283 . . . 4 GrpOpHom
1110simprd 450 . . 3 GrpOpHom
12 ffvelrn 5868 . . . . . . . 8
13 ffvelrn 5868 . . . . . . . 8
1412, 13anim12dan 811 . . . . . . 7
157, 14sylan 458 . . . . . 6 GrpOpHom
162, 3ghomgrp 25101 . . . . . . 7 GrpOpHom
174subgoov 21893 . . . . . . 7
1816, 17sylan 458 . . . . . 6 GrpOpHom
1915, 18syldan 457 . . . . 5 GrpOpHom
2019eqeq1d 2444 . . . 4 GrpOpHom
21202ralbidva 2745 . . 3 GrpOpHom
2211, 21mpbird 224 . 2 GrpOpHom
23 issubgo 21891 . . . . 5
2416, 23sylib 189 . . . 4 GrpOpHom
2524simp2d 970 . . 3 GrpOpHom
261, 4elghom 21951 . . . . 5 GrpOpHom
2726biimprd 215 . . . 4 GrpOpHom
28273adant3 977 . . 3 GrpOpHom GrpOpHom
2925, 28syld3an2 1231 . 2 GrpOpHom GrpOpHom
307, 22, 29mp2and 661 1 GrpOpHom GrpOpHom
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705   wss 3320   cxp 4876   crn 4879   cres 4880  wf 5450  wfo 5452  cfv 5454  (class class class)co 6081  cgr 21774  csubgo 21889   GrpOpHom cghom 21945 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-grpo 21779  df-gid 21780  df-ginv 21781  df-subgo 21890  df-ghom 21946
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