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Theorem ghomfo 25094
 Description: A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
Hypotheses
Ref Expression
ghomfo.1
ghomfo.2
ghomfo.3
ghomfo.4
Assertion
Ref Expression
ghomfo GrpOpHom

Proof of Theorem ghomfo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ghomfo.1 . . . . . 6
2 eqid 2435 . . . . . 6
31, 2elghom 21943 . . . . 5 GrpOpHom
43biimp3a 1283 . . . 4 GrpOpHom
54simpld 446 . . 3 GrpOpHom
6 ffn 5583 . . 3
75, 6syl 16 . 2 GrpOpHom
8 ghomfo.3 . . . . . 6
98dmeqi 5063 . . . . 5
10 ghomfo.2 . . . . . . . . 9
1110, 8ghomgrp 25093 . . . . . . . 8 GrpOpHom
12 issubgo 21883 . . . . . . . 8
1311, 12sylib 189 . . . . . . 7 GrpOpHom
1413simp2d 970 . . . . . 6 GrpOpHom
15 ghomfo.4 . . . . . . . 8
1615grpofo 21779 . . . . . . 7
17 fof 5645 . . . . . . 7
18 fdm 5587 . . . . . . 7
1916, 17, 183syl 19 . . . . . 6
2014, 19syl 16 . . . . 5 GrpOpHom
21 frn 5589 . . . . . . . . 9
225, 21syl 16 . . . . . . . 8 GrpOpHom
2310, 22syl5eqss 3384 . . . . . . 7 GrpOpHom
24 xpss12 4973 . . . . . . 7
2523, 23, 24syl2anc 643 . . . . . 6 GrpOpHom
26 ssdmres 5160 . . . . . . . 8
272grpofo 21779 . . . . . . . . . 10
28 fof 5645 . . . . . . . . . 10
29 fdm 5587 . . . . . . . . . 10
3027, 28, 293syl 19 . . . . . . . . 9
3130sseq2d 3368 . . . . . . . 8
3226, 31syl5rbbr 252 . . . . . . 7
33323ad2ant2 979 . . . . . 6 GrpOpHom
3425, 33mpbid 202 . . . . 5 GrpOpHom
359, 20, 343eqtr3a 2491 . . . 4 GrpOpHom
36 xpid11 5083 . . . 4
3735, 36sylib 189 . . 3 GrpOpHom
3837, 10syl6req 2484 . 2 GrpOpHom
39 df-fo 5452 . 2
407, 38, 39sylanbrc 646 1 GrpOpHom
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697   wss 3312   cxp 4868   cdm 4870   crn 4871   cres 4872   wfn 5441  wf 5442  wfo 5444  cfv 5446  (class class class)co 6073  cgr 21766  csubgo 21881   GrpOpHom cghom 21937 This theorem is referenced by:  ghomcl  25095  ghomgsg  25096  ghomf1olem  25097 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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-grpo 21771  df-gid 21772  df-ginv 21773  df-subgo 21882  df-ghom 21938
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