Theorem homafval 16679

Description: Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
homarcl.h	|- H = (Homa ` C )
homafval.b	|- B = ( Base ` C )
homafval.c	|- ( ph -> C e. Cat )
homafval.j	|- J = ( Hom ` C )

Assertion

Ref	Expression
homafval	|- ( ph -> H = ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) )

Distinct variable groups:   x, B   x, C   ph, x

Allowed substitution hints:   H( x)   J( x)

Proof of Theorem homafval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		homarcl.h	. 2 |- H = (Homa ` C )
2		homafval.c	. . 3 |- ( ph -> C e. Cat )
3		fveq2 6191	. . . . . . 7 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
4		homafval.b	. . . . . . 7 |- B = ( Base ` C )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( c = C -> ( Base ` c ) = B )
6	5	sqxpeqd 5141	. . . . 5 |- ( c = C -> ( ( Base ` c ) X. ( Base ` c ) ) = ( B X. B ) )
7		fveq2 6191	. . . . . . . 8 |- ( c = C -> ( Hom ` c ) = ( Hom ` C ) )
8		homafval.j	. . . . . . . 8 |- J = ( Hom ` C )
9	7, 8	syl6eqr 2674	. . . . . . 7 |- ( c = C -> ( Hom ` c ) = J )
10	9	fveq1d 6193	. . . . . 6 |- ( c = C -> ( ( Hom ` c ) ` x ) = ( J ` x ) )
11	10	xpeq2d 5139	. . . . 5 |- ( c = C -> ( { x } X. ( ( Hom ` c ) ` x ) ) = ( { x } X. ( J ` x ) ) )
12	6, 11	mpteq12dv 4733	. . . 4 |- ( c = C -> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) |-> ( { x } X. ( ( Hom ` c ) ` x ) ) ) = ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) )
13		df-homa 16676	. . . 4 |- Homa = ( c e. Cat |-> ( x e. ( ( Base ` c ) X. ( Base ` c ) ) |-> ( { x } X. ( ( Hom ` c ) ` x ) ) ) )
14		fvex 6201	. . . . . . 7 |- ( Base ` C ) e. _V
15	4, 14	eqeltri 2697	. . . . . 6 |- B e. _V
16	15, 15	xpex 6962	. . . . 5 |- ( B X. B ) e. _V
17	16	mptex 6486	. . . 4 |- ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) e. _V
18	12, 13, 17	fvmpt 6282	. . 3 |- ( C e. Cat -> (Homa ` C ) = ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) )
19	2, 18	syl 17	. 2 |- ( ph -> (Homa ` C ) = ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) )
20	1, 19	syl5eq 2668	1 |- ( ph -> H = ( x e. ( B X. B ) |-> ( { x } X. ( J ` x ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   |-> cmpt 4729   X. cxp 5112   ` cfv 5888   Basecbs 15857   Hom chom 15952   Catccat 16325  Hom_achoma 16673

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-homa 16676

This theorem is referenced by:  homaf  16680  homaval  16681