Theorem fulli 16573

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

Step	Hyp	Ref	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 )
7	1, 2, 3, 4, 5, 6	fullfo 16572	. 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 6378	. 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 ) )
10	7, 8, 9	syl2anc 693	1 |- ( ph -> E. f e. ( X H Y ) R = ( ( X G Y ) ` f ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Full cful 16562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-func 16518  df-full 16564

This theorem is referenced by:  ffthiso  16589