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Theorem fulli 13836
Description: The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
isfull.b  |-  B  =  ( Base `  C
)
isfull.j  |-  J  =  (  Hom  `  D
)
isfull.h  |-  H  =  (  Hom  `  C
)
fullfo.f  |-  ( ph  ->  F ( C Full  D
) G )
fullfo.x  |-  ( ph  ->  X  e.  B )
fullfo.y  |-  ( ph  ->  Y  e.  B )
fulli.r  |-  ( ph  ->  R  e.  ( ( F `  X ) J ( F `  Y ) ) )
Assertion
Ref Expression
fulli  |-  ( ph  ->  E. f  e.  ( X H Y ) R  =  ( ( X G Y ) `
 f ) )
Distinct variable groups:    B, f    C, f    D, f    f, H   
f, J    R, f    f, X    f, Y    f, F    f, G
Allowed substitution hint:    ph( f)

Proof of Theorem fulli
StepHypRef Expression
1 isfull.b . . 3  |-  B  =  ( Base `  C
)
2 isfull.j . . 3  |-  J  =  (  Hom  `  D
)
3 isfull.h . . 3  |-  H  =  (  Hom  `  C
)
4 fullfo.f . . 3  |-  ( ph  ->  F ( C Full  D
) G )
5 fullfo.x . . 3  |-  ( ph  ->  X  e.  B )
6 fullfo.y . . 3  |-  ( ph  ->  Y  e.  B )
71, 2, 3, 4, 5, 6fullfo 13835 . 2  |-  ( ph  ->  ( X G Y ) : ( X H Y ) -onto-> ( ( F `  X
) J ( F `
 Y ) ) )
8 fulli.r . 2  |-  ( ph  ->  R  e.  ( ( F `  X ) J ( F `  Y ) ) )
9 foelrn 5717 . 2  |-  ( ( ( X G Y ) : ( X H Y ) -onto-> ( ( F `  X
) J ( F `
 Y ) )  /\  R  e.  ( ( F `  X
) J ( F `
 Y ) ) )  ->  E. f  e.  ( X H Y ) R  =  ( ( X G Y ) `  f ) )
107, 8, 9syl2anc 642 1  |-  ( ph  ->  E. f  e.  ( X H Y ) R  =  ( ( X G Y ) `
 f ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1633    e. wcel 1701   E.wrex 2578   class class class wbr 4060   -onto->wfo 5290   ` cfv 5292  (class class class)co 5900   Basecbs 13195    Hom chom 13266   Full cful 13825
This theorem is referenced by:  ffthiso  13852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-map 6817  df-ixp 6861  df-func 13781  df-full 13827
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