Theorem fthf1 16577

Description: The morphism map of a faithful functor is an injection. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
isfth.b	|- B = ( Base ` C )
isfth.h	|- H = ( Hom ` C )
isfth.j	|- J = ( Hom ` D )
fthf1.f	|- ( ph -> F ( C Faith D ) G )
fthf1.x	|- ( ph -> X e. B )
fthf1.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
fthf1	|- ( ph -> ( X G Y ) : ( X H Y ) -1-1-> ( ( F ` X ) J ( F ` Y ) ) )

Proof of Theorem fthf1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fthf1.f	. . 3 |- ( ph -> F ( C Faith D ) G )
2		isfth.b	. . . . 5 |- B = ( Base ` C )
3		isfth.h	. . . . 5 |- H = ( Hom ` C )
4		isfth.j	. . . . 5 |- J = ( Hom ` D )
5	2, 3, 4	isfth2 16575	. . . 4 |- ( F ( C Faith D ) G <-> ( F ( C Func D ) G /\ A. x e. B A. y e. B ( x G y ) : ( x H y ) -1-1-> ( ( F ` x ) J ( F ` y ) ) ) )
6	5	simprbi 480	. . 3 |- ( F ( C Faith D ) G -> A. x e. B A. y e. B ( x G y ) : ( x H y ) -1-1-> ( ( F ` x ) J ( F ` y ) ) )
7	1, 6	syl 17	. 2 |- ( ph -> A. x e. B A. y e. B ( x G y ) : ( x H y ) -1-1-> ( ( F ` x ) J ( F ` y ) ) )
8		fthf1.x	. . 3 |- ( ph -> X e. B )
9		fthf1.y	. . . . 5 |- ( ph -> Y e. B )
10	9	adantr 481	. . . 4 |- ( ( ph /\ x = X ) -> Y e. B )
11		simplr 792	. . . . . 6 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> x = X )
12		simpr 477	. . . . . 6 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> y = Y )
13	11, 12	oveq12d 6668	. . . . 5 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> ( x G y ) = ( X G Y ) )
14	11, 12	oveq12d 6668	. . . . 5 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> ( x H y ) = ( X H Y ) )
15	11	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> ( F ` x ) = ( F ` X ) )
16	12	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> ( F ` y ) = ( F ` Y ) )
17	15, 16	oveq12d 6668	. . . . 5 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> ( ( F ` x ) J ( F ` y ) ) = ( ( F ` X ) J ( F ` Y ) ) )
18	13, 14, 17	f1eq123d 6131	. . . 4 |- ( ( ( ph /\ x = X ) /\ y = Y ) -> ( ( x G y ) : ( x H y ) -1-1-> ( ( F ` x ) J ( F ` y ) ) <-> ( X G Y ) : ( X H Y ) -1-1-> ( ( F ` X ) J ( F ` Y ) ) ) )
19	10, 18	rspcdv 3312	. . 3 |- ( ( ph /\ x = X ) -> ( A. y e. B ( x G y ) : ( x H y ) -1-1-> ( ( F ` x ) J ( F ` y ) ) -> ( X G Y ) : ( X H Y ) -1-1-> ( ( F ` X ) J ( F ` Y ) ) ) )
20	8, 19	rspcimdv 3310	. 2 |- ( ph -> ( A. x e. B A. y e. B ( x G y ) : ( x H y ) -1-1-> ( ( F ` x ) J ( F ` y ) ) -> ( X G Y ) : ( X H Y ) -1-1-> ( ( F ` X ) J ( F ` Y ) ) ) )
21	7, 20	mpd 15	1 |- ( ph -> ( X G Y ) : ( X H Y ) -1-1-> ( ( F ` X ) J ( F ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Func cfunc 16514   Faith cfth 16563

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-fth 16565

This theorem is referenced by:  fthi  16578  ffthf1o  16579  fthoppc  16583  cofth  16595