Theorem zerooval 16649

Description: The value of the zero object function, i.e. the set of all zero 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
zerooval	|- ( ph -> (ZeroO ` C ) = ( (InitO ` C ) i^i (TermO ` C ) ) )

Proof of Theorem zerooval

Dummy variable c is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-zeroo 16643	. . 3 |- ZeroO = ( c e. Cat |-> ( (InitO ` c ) i^i (TermO ` c ) ) )
2	1	a1i 11	. 2 |- ( ph -> ZeroO = ( c e. Cat |-> ( (InitO ` c ) i^i (TermO ` c ) ) ) )
3		fveq2 6191	. . . 4 |- ( c = C -> (InitO ` c ) = (InitO ` C ) )
4		fveq2 6191	. . . 4 |- ( c = C -> (TermO ` c ) = (TermO ` C ) )
5	3, 4	ineq12d 3815	. . 3 |- ( c = C -> ( (InitO ` c ) i^i (TermO ` c ) ) = ( (InitO ` C ) i^i (TermO ` C ) ) )
6	5	adantl 482	. 2 |- ( ( ph /\ c = C ) -> ( (InitO ` c ) i^i (TermO ` c ) ) = ( (InitO ` C ) i^i (TermO ` C ) ) )
7		initoval.c	. 2 |- ( ph -> C e. Cat )
8		fvex 6201	. . . 4 |- (InitO ` C ) e. _V
9	8	inex1 4799	. . 3 |- ( (InitO ` C ) i^i (TermO ` C ) ) e. _V
10	9	a1i 11	. 2 |- ( ph -> ( (InitO ` C ) i^i (TermO ` C ) ) e. _V )
11	2, 6, 7, 10	fvmptd 6288	1 |- ( ph -> (ZeroO ` C ) = ( (InitO ` C ) i^i (TermO ` C ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   |-> cmpt 4729   ` cfv 5888   Basecbs 15857   Hom chom 15952   Catccat 16325  InitOcinito 16638  TermOctermo 16639  ZeroOczeroo 16640

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-zeroo 16643

This theorem is referenced by:  iszeroo  16652  iszeroi  16659