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Theorem mgmhmf 32863
Description: A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.)
Hypotheses
Ref Expression
mgmhmf.b  |-  B  =  ( Base `  S
)
mgmhmf.c  |-  C  =  ( Base `  T
)
Assertion
Ref Expression
mgmhmf  |-  ( F  e.  ( S MgmHom  T
)  ->  F : B
--> C )

Proof of Theorem mgmhmf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmf.b . . 3  |-  B  =  ( Base `  S
)
2 mgmhmf.c . . 3  |-  C  =  ( Base `  T
)
3 eqid 2454 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
4 eqid 2454 . . 3  |-  ( +g  `  T )  =  ( +g  `  T )
51, 2, 3, 4ismgmhm 32862 . 2  |-  ( F  e.  ( S MgmHom  T
)  <->  ( ( S  e. Mgm  /\  T  e. Mgm )  /\  ( F : B
--> C  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x ( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T
) ( F `  y ) ) ) ) )
6 simprl 754 . 2  |-  ( ( ( S  e. Mgm  /\  T  e. Mgm )  /\  ( F : B --> C  /\  A. x  e.  B  A. y  e.  B  ( F `  ( x
( +g  `  S ) y ) )  =  ( ( F `  x ) ( +g  `  T ) ( F `
 y ) ) ) )  ->  F : B --> C )
75, 6sylbi 195 1  |-  ( F  e.  ( S MgmHom  T
)  ->  F : B
--> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   -->wf 5566   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787  Mgmcmgm 16072   MgmHom cmgmhm 32856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-mgmhm 32858
This theorem is referenced by:  mgmhmf1o  32866  resmgmhm  32877  resmgmhm2  32878  resmgmhm2b  32879  mgmhmco  32880  mgmhmima  32881  mgmhmeql  32882
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