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Theorem fulli 13787
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 13786 . 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 5679 . 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 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219   Full cful 13776
This theorem is referenced by:  ffthiso  13803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-ixp 6818  df-func 13732  df-full 13778
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