Theorem initoval 16647

Description: The value of the initial object function, i.e. the set of all initial objects of a category. (Contributed by AV, 3-Apr-2020.)

Hypotheses

Ref	Expression
initoval.c	|- ( ph -> C e. Cat )
initoval.b	|- B = ( Base ` C )
initoval.h	|- H = ( Hom ` C )

Assertion

Ref	Expression
initoval	|- ( ph -> (InitO ` C ) = { a e. B | A. b e. B E! h h e. ( a H b ) } )

Distinct variable groups:   a, b, h   B, a, b   C, a, b, h

Allowed substitution hints:   ph( h, a, b)   B( h)   H( h, a, b)

Proof of Theorem initoval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-inito 16641	. . 3 |- InitO = ( c e. Cat |-> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } )
2	1	a1i 11	. 2 |- ( ph -> InitO = ( c e. Cat |-> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } ) )
3		fveq2 6191	. . . . 5 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
4		initoval.b	. . . . 5 |- B = ( Base ` C )
5	3, 4	syl6eqr 2674	. . . 4 |- ( c = C -> ( Base ` c ) = B )
6		fveq2 6191	. . . . . . . . 9 |- ( c = C -> ( Hom ` c ) = ( Hom ` C ) )
7		initoval.h	. . . . . . . . 9 |- H = ( Hom ` C )
8	6, 7	syl6eqr 2674	. . . . . . . 8 |- ( c = C -> ( Hom ` c ) = H )
9	8	oveqd 6667	. . . . . . 7 |- ( c = C -> ( a ( Hom ` c ) b ) = ( a H b ) )
10	9	eleq2d 2687	. . . . . 6 |- ( c = C -> ( h e. ( a ( Hom ` c ) b ) <-> h e. ( a H b ) ) )
11	10	eubidv 2490	. . . . 5 |- ( c = C -> ( E! h h e. ( a ( Hom ` c ) b ) <-> E! h h e. ( a H b ) ) )
12	5, 11	raleqbidv 3152	. . . 4 |- ( c = C -> ( A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) <-> A. b e. B E! h h e. ( a H b ) ) )
13	5, 12	rabeqbidv 3195	. . 3 |- ( c = C -> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } = { a e. B | A. b e. B E! h h e. ( a H b ) } )
14	13	adantl 482	. 2 |- ( ( ph /\ c = C ) -> { a e. ( Base ` c ) | A. b e. ( Base ` c ) E! h h e. ( a ( Hom ` c ) b ) } = { a e. B | A. b e. B E! h h e. ( a H b ) } )
15		initoval.c	. 2 |- ( ph -> C e. Cat )
16		fvex 6201	. . . . 5 |- ( Base ` C ) e. _V
17	4, 16	eqeltri 2697	. . . 4 |- B e. _V
18	17	rabex 4813	. . 3 |- { a e. B | A. b e. B E! h h e. ( a H b ) } e. _V
19	18	a1i 11	. 2 |- ( ph -> { a e. B | A. b e. B E! h h e. ( a H b ) } e. _V )
20	2, 14, 15, 19	fvmptd 6288	1 |- ( ph -> (InitO ` C ) = { a e. B | A. b e. B E! h h e. ( a H b ) } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   E!weu 2470   A.wral 2912   {crab 2916   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Catccat 16325  InitOcinito 16638

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-inito 16641

This theorem is referenced by:  isinito  16650  isinitoi  16653