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Theorem fulli 14112
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 14111 . 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 5890 . 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 644 1  |-  ( ph  ->  E. f  e.  ( X H Y ) R  =  ( ( X G Y ) `
 f ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214   -onto->wfo 5454   ` cfv 5456  (class class class)co 6083   Basecbs 13471    Hom chom 13542   Full cful 14101
This theorem is referenced by:  ffthiso  14128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-map 7022  df-ixp 7066  df-func 14057  df-full 14103
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