Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elghom Structured version   Unicode version

Theorem elghom 21943
 Description: Membership in the set of group homomorphisms from to . (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
elghom.1
elghom.2
Assertion
Ref Expression
elghom GrpOpHom
Distinct variable groups:   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem elghom
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . 3
21elghomlem2 21942 . 2 GrpOpHom
3 elghom.1 . . . 4
4 elghom.2 . . . 4
53, 4feq23i 5579 . . 3
63raleqi 2900 . . . 4
73, 6raleqbii 2727 . . 3
85, 7anbi12i 679 . 2
92, 8syl6bbr 255 1 GrpOpHom
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cab 2421  wral 2697   crn 4871  wf 5442  cfv 5446  (class class class)co 6073  cgr 21766   GrpOpHom cghom 21937 This theorem is referenced by:  ghomlin  21944  ghomid  21945  ghomgrpilem1  25088  ghomgrpilem2  25089  ghomsn  25091  ghomfo  25094  ghomgsg  25096  ghomf  26538  ghomco  26539  rngogrphom  26568 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-ghom 21938
 Copyright terms: Public domain W3C validator