Theorem sectcan 16415

Description: If G is a section of F and F is a section of H, then G = H. Proposition 3.10 of [Adamek] p. 28. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
sectcan.b	|- B = ( Base ` C )
sectcan.s	|- S = (Sect ` C )
sectcan.c	|- ( ph -> C e. Cat )
sectcan.x	|- ( ph -> X e. B )
sectcan.y	|- ( ph -> Y e. B )
sectcan.1	|- ( ph -> G ( X S Y ) F )
sectcan.2	|- ( ph -> F ( Y S X ) H )

Assertion

Ref	Expression
sectcan	|- ( ph -> G = H )

Proof of Theorem sectcan

Step	Hyp	Ref	Expression
1		sectcan.b	. . . 4 |- B = ( Base ` C )
2		eqid 2622	. . . 4 |- ( Hom ` C ) = ( Hom ` C )
3		eqid 2622	. . . 4 |- (comp ` C ) = (comp ` C )
4		sectcan.c	. . . 4 |- ( ph -> C e. Cat )
5		sectcan.x	. . . 4 |- ( ph -> X e. B )
6		sectcan.y	. . . 4 |- ( ph -> Y e. B )
7		sectcan.1	. . . . . 6 |- ( ph -> G ( X S Y ) F )
8		eqid 2622	. . . . . . 7 |- ( Id ` C ) = ( Id ` C )
9		sectcan.s	. . . . . . 7 |- S = (Sect ` C )
10	1, 2, 3, 8, 9, 4, 5, 6	issect 16413	. . . . . 6 |- ( ph -> ( G ( X S Y ) F <-> ( G e. ( X ( Hom ` C ) Y ) /\ F e. ( Y ( Hom ` C ) X ) /\ ( F ( <. X , Y >. (comp ` C ) X ) G ) = ( ( Id ` C ) ` X ) ) ) )
11	7, 10	mpbid 222	. . . . 5 |- ( ph -> ( G e. ( X ( Hom ` C ) Y ) /\ F e. ( Y ( Hom ` C ) X ) /\ ( F ( <. X , Y >. (comp ` C ) X ) G ) = ( ( Id ` C ) ` X ) ) )
12	11	simp1d 1073	. . . 4 |- ( ph -> G e. ( X ( Hom ` C ) Y ) )
13		sectcan.2	. . . . . 6 |- ( ph -> F ( Y S X ) H )
14	1, 2, 3, 8, 9, 4, 6, 5	issect 16413	. . . . . 6 |- ( ph -> ( F ( Y S X ) H <-> ( F e. ( Y ( Hom ` C ) X ) /\ H e. ( X ( Hom ` C ) Y ) /\ ( H ( <. Y , X >. (comp ` C ) Y ) F ) = ( ( Id ` C ) ` Y ) ) ) )
15	13, 14	mpbid 222	. . . . 5 |- ( ph -> ( F e. ( Y ( Hom ` C ) X ) /\ H e. ( X ( Hom ` C ) Y ) /\ ( H ( <. Y , X >. (comp ` C ) Y ) F ) = ( ( Id ` C ) ` Y ) ) )
16	15	simp1d 1073	. . . 4 |- ( ph -> F e. ( Y ( Hom ` C ) X ) )
17	15	simp2d 1074	. . . 4 |- ( ph -> H e. ( X ( Hom ` C ) Y ) )
18	1, 2, 3, 4, 5, 6, 5, 12, 16, 6, 17	catass 16347	. . 3 |- ( ph -> ( ( H ( <. Y , X >. (comp ` C ) Y ) F ) ( <. X , Y >. (comp ` C ) Y ) G ) = ( H ( <. X , X >. (comp ` C ) Y ) ( F ( <. X , Y >. (comp ` C ) X ) G ) ) )
19	15	simp3d 1075	. . . 4 |- ( ph -> ( H ( <. Y , X >. (comp ` C ) Y ) F ) = ( ( Id ` C ) ` Y ) )
20	19	oveq1d 6665	. . 3 |- ( ph -> ( ( H ( <. Y , X >. (comp ` C ) Y ) F ) ( <. X , Y >. (comp ` C ) Y ) G ) = ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) G ) )
21	11	simp3d 1075	. . . 4 |- ( ph -> ( F ( <. X , Y >. (comp ` C ) X ) G ) = ( ( Id ` C ) ` X ) )
22	21	oveq2d 6666	. . 3 |- ( ph -> ( H ( <. X , X >. (comp ` C ) Y ) ( F ( <. X , Y >. (comp ` C ) X ) G ) ) = ( H ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) )
23	18, 20, 22	3eqtr3d 2664	. 2 |- ( ph -> ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) G ) = ( H ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) )
24	1, 2, 8, 4, 5, 3, 6, 12	catlid 16344	. 2 |- ( ph -> ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) G ) = G )
25	1, 2, 8, 4, 5, 3, 6, 17	catrid 16345	. 2 |- ( ph -> ( H ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) = H )
26	23, 24, 25	3eqtr3d 2664	1 |- ( ph -> G = H )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  Sectcsect 16404

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-rmo 2920  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cat 16329  df-cid 16330  df-sect 16407

This theorem is referenced by:  invfun  16424  inveq  16434