Theorem oppcmon 16398

Description: A monomorphism in the opposite category is an epimorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
oppcmon.o	|- O = (oppCat ` C )
oppcmon.c	|- ( ph -> C e. Cat )
oppcmon.m	|- M = (Mono ` O )
oppcmon.e	|- E = (Epi ` C )

Assertion

Ref	Expression
oppcmon	|- ( ph -> ( X M Y ) = ( Y E X ) )

Proof of Theorem oppcmon

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppcmon.e	. . . 4 |- E = (Epi ` C )
2		oppcmon.c	. . . . 5 |- ( ph -> C e. Cat )
3		fveq2 6191	. . . . . . . . . 10 |- ( c = C -> (oppCat ` c ) = (oppCat ` C ) )
4		oppcmon.o	. . . . . . . . . 10 |- O = (oppCat ` C )
5	3, 4	syl6eqr 2674	. . . . . . . . 9 |- ( c = C -> (oppCat ` c ) = O )
6	5	fveq2d 6195	. . . . . . . 8 |- ( c = C -> (Mono ` (oppCat ` c ) ) = (Mono ` O ) )
7		oppcmon.m	. . . . . . . 8 |- M = (Mono ` O )
8	6, 7	syl6eqr 2674	. . . . . . 7 |- ( c = C -> (Mono ` (oppCat ` c ) ) = M )
9	8	tposeqd 7355	. . . . . 6 |- ( c = C -> tpos (Mono ` (oppCat ` c ) ) = tpos M )
10		df-epi 16391	. . . . . 6 |- Epi = ( c e. Cat |-> tpos (Mono ` (oppCat ` c ) ) )
11		fvex 6201	. . . . . . . 8 |- (Mono ` O ) e. _V
12	7, 11	eqeltri 2697	. . . . . . 7 |- M e. _V
13	12	tposex 7386	. . . . . 6 |- tpos M e. _V
14	9, 10, 13	fvmpt 6282	. . . . 5 |- ( C e. Cat -> (Epi ` C ) = tpos M )
15	2, 14	syl 17	. . . 4 |- ( ph -> (Epi ` C ) = tpos M )
16	1, 15	syl5eq 2668	. . 3 |- ( ph -> E = tpos M )
17	16	oveqd 6667	. 2 |- ( ph -> ( Y E X ) = ( Ytpos M X ) )
18		ovtpos 7367	. 2 |- ( Ytpos M X ) = ( X M Y )
19	17, 18	syl6req 2673	1 |- ( ph -> ( X M Y ) = ( Y E X ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ` cfv 5888  (class class class)co 6650  tpos ctpos 7351   Catccat 16325  oppCatcoppc 16371  Monocmon 16388  Epicepi 16389

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-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-rab 2921  df-v 3202  df-sbc 3436  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-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-fv 5896  df-ov 6653  df-tpos 7352  df-epi 16391

This theorem is referenced by:  oppcepi  16399  isepi  16400  epii  16403  sectepi  16444  episect  16445  fthepi  16588