Theorem 1stf1 16832

Description: Value of the first projection on an object. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
1stfval.t	|- T = ( C X.c D )
1stfval.b	|- B = ( Base ` T )
1stfval.h	|- H = ( Hom ` T )
1stfval.c	|- ( ph -> C e. Cat )
1stfval.d	|- ( ph -> D e. Cat )
1stfval.p	|- P = ( C 1stF D )
1stf1.p	|- ( ph -> R e. B )

Assertion

Ref	Expression
1stf1	|- ( ph -> ( ( 1st ` P ) ` R ) = ( 1st ` R ) )

Proof of Theorem 1stf1

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

Step	Hyp	Ref	Expression
1		1stfval.t	. . . . 5 |- T = ( C X.c D )
2		1stfval.b	. . . . 5 |- B = ( Base ` T )
3		1stfval.h	. . . . 5 |- H = ( Hom ` T )
4		1stfval.c	. . . . 5 |- ( ph -> C e. Cat )
5		1stfval.d	. . . . 5 |- ( ph -> D e. Cat )
6		1stfval.p	. . . . 5 |- P = ( C 1stF D )
7	1, 2, 3, 4, 5, 6	1stfval 16831	. . . 4 |- ( ph -> P = <. ( 1st |` B ) , ( x e. B , y e. B |-> ( 1st |` ( x H y ) ) ) >. )
8		fo1st 7188	. . . . . . 7 |- 1st : _V -onto-> _V
9		fofun 6116	. . . . . . 7 |- ( 1st : _V -onto-> _V -> Fun 1st )
10	8, 9	ax-mp 5	. . . . . 6 |- Fun 1st
11		fvex 6201	. . . . . . 7 |- ( Base ` T ) e. _V
12	2, 11	eqeltri 2697	. . . . . 6 |- B e. _V
13		resfunexg 6479	. . . . . 6 |- ( ( Fun 1st /\ B e. _V ) -> ( 1st |` B ) e. _V )
14	10, 12, 13	mp2an 708	. . . . 5 |- ( 1st |` B ) e. _V
15	12, 12	mpt2ex 7247	. . . . 5 |- ( x e. B , y e. B |-> ( 1st |` ( x H y ) ) ) e. _V
16	14, 15	op1std 7178	. . . 4 |- ( P = <. ( 1st |` B ) , ( x e. B , y e. B |-> ( 1st |` ( x H y ) ) ) >. -> ( 1st ` P ) = ( 1st |` B ) )
17	7, 16	syl 17	. . 3 |- ( ph -> ( 1st ` P ) = ( 1st |` B ) )
18	17	fveq1d 6193	. 2 |- ( ph -> ( ( 1st ` P ) ` R ) = ( ( 1st |` B ) ` R ) )
19		1stf1.p	. . 3 |- ( ph -> R e. B )
20		fvres 6207	. . 3 |- ( R e. B -> ( ( 1st |` B ) ` R ) = ( 1st ` R ) )
21	19, 20	syl 17	. 2 |- ( ph -> ( ( 1st |` B ) ` R ) = ( 1st ` R ) )
22	18, 21	eqtrd 2656	1 |- ( ph -> ( ( 1st ` P ) ` R ) = ( 1st ` R ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   |` cres 5116   Fun wfun 5882   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   Basecbs 15857   Hom chom 15952   Catccat 16325   X._c cxpc 16808   1st_F c1stf 16809

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-xpc 16812  df-1stf 16813

This theorem is referenced by:  prf1st  16844  1st2ndprf  16846  uncf1  16876  uncf2  16877  diag11  16883  yonedalem21  16913  yonedalem22  16918