Theorem homffval 16350

Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
homffval.f	|- F = ( Hom f ` C )
homffval.b	|- B = ( Base ` C )
homffval.h	|- H = ( Hom ` C )

Assertion

Ref	Expression
homffval	|- F = ( x e. B , y e. B |-> ( x H y ) )

Distinct variable groups:   x, y, B   x, C, y   x, H, y

Allowed substitution hints:   F( x, y)

Proof of Theorem homffval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		homffval.f	. 2 |- F = ( Hom f ` C )
2		fveq2 6191	. . . . . 6 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
3		homffval.b	. . . . . 6 |- B = ( Base ` C )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( c = C -> ( Base ` c ) = B )
5		fveq2 6191	. . . . . . 7 |- ( c = C -> ( Hom ` c ) = ( Hom ` C ) )
6		homffval.h	. . . . . . 7 |- H = ( Hom ` C )
7	5, 6	syl6eqr 2674	. . . . . 6 |- ( c = C -> ( Hom ` c ) = H )
8	7	oveqd 6667	. . . . 5 |- ( c = C -> ( x ( Hom ` c ) y ) = ( x H y ) )
9	4, 4, 8	mpt2eq123dv 6717	. . . 4 |- ( c = C -> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( x ( Hom ` c ) y ) ) = ( x e. B , y e. B |-> ( x H y ) ) )
10		df-homf 16331	. . . 4 |- Hom f = ( c e. _V |-> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> ( x ( Hom ` c ) y ) ) )
11		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
12	3, 11	eqeltri 2697	. . . . 5 |- B e. _V
13	12, 12	mpt2ex 7247	. . . 4 |- ( x e. B , y e. B |-> ( x H y ) ) e. _V
14	9, 10, 13	fvmpt 6282	. . 3 |- ( C e. _V -> ( Hom f ` C ) = ( x e. B , y e. B |-> ( x H y ) ) )
15		mpt20 6725	. . . . 5 |- ( x e. (/) , y e. (/) |-> ( x H y ) ) = (/)
16	15	eqcomi 2631	. . . 4 |- (/) = ( x e. (/) , y e. (/) |-> ( x H y ) )
17		fvprc 6185	. . . 4 |- ( -. C e. _V -> ( Hom f ` C ) = (/) )
18		fvprc 6185	. . . . . 6 |- ( -. C e. _V -> ( Base ` C ) = (/) )
19	3, 18	syl5eq 2668	. . . . 5 |- ( -. C e. _V -> B = (/) )
20		mpt2eq12 6715	. . . . 5 |- ( ( B = (/) /\ B = (/) ) -> ( x e. B , y e. B |-> ( x H y ) ) = ( x e. (/) , y e. (/) |-> ( x H y ) ) )
21	19, 19, 20	syl2anc 693	. . . 4 |- ( -. C e. _V -> ( x e. B , y e. B |-> ( x H y ) ) = ( x e. (/) , y e. (/) |-> ( x H y ) ) )
22	16, 17, 21	3eqtr4a 2682	. . 3 |- ( -. C e. _V -> ( Hom f ` C ) = ( x e. B , y e. B |-> ( x H y ) ) )
23	14, 22	pm2.61i 176	. 2 |- ( Hom f ` C ) = ( x e. B , y e. B |-> ( x H y ) )
24	1, 23	eqtri 2644	1 |- F = ( x e. B , y e. B |-> ( x H y ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   Hom chom 15952   Hom _f chomf 16327

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-homf 16331

This theorem is referenced by:  fnhomeqhomf  16351  homfval  16352  homffn  16353  homfeq  16354  oppchomf  16380  reschomf  16491