Theorem initoid 16655

Description: For an initial object, the identity arrow is the one and only morphism of the object to the object itself. (Contributed by AV, 6-Apr-2020.)

Hypotheses

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

Assertion

Ref	Expression
initoid	|- ( ( ph /\ O e. (InitO ` C ) ) -> ( O H O ) = { ( ( Id ` C ) ` O ) } )

Proof of Theorem initoid

Dummy variables h o are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isinitoi.b	. . 3 |- B = ( Base ` C )
2		isinitoi.h	. . 3 |- H = ( Hom ` C )
3		isinitoi.c	. . 3 |- ( ph -> C e. Cat )
4	1, 2, 3	isinitoi 16653	. 2 |- ( ( ph /\ O e. (InitO ` C ) ) -> ( O e. B /\ A. o e. B E! h h e. ( O H o ) ) )
5		oveq2 6658	. . . . . . . 8 |- ( o = O -> ( O H o ) = ( O H O ) )
6	5	eleq2d 2687	. . . . . . 7 |- ( o = O -> ( h e. ( O H o ) <-> h e. ( O H O ) ) )
7	6	eubidv 2490	. . . . . 6 |- ( o = O -> ( E! h h e. ( O H o ) <-> E! h h e. ( O H O ) ) )
8	7	rspcv 3305	. . . . 5 |- ( O e. B -> ( A. o e. B E! h h e. ( O H o ) -> E! h h e. ( O H O ) ) )
9	8	adantl 482	. . . 4 |- ( ( ( ph /\ O e. (InitO ` C ) ) /\ O e. B ) -> ( A. o e. B E! h h e. ( O H o ) -> E! h h e. ( O H O ) ) )
10		eusn 4265	. . . . 5 |- ( E! h h e. ( O H O ) <-> E. h ( O H O ) = { h } )
11		eqid 2622	. . . . . . . . 9 |- ( Id ` C ) = ( Id ` C )
12	3	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ O e. (InitO ` C ) ) /\ O e. B ) -> C e. Cat )
13		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ O e. (InitO ` C ) ) /\ O e. B ) -> O e. B )
14	1, 2, 11, 12, 13	catidcl 16343	. . . . . . . 8 |- ( ( ( ph /\ O e. (InitO ` C ) ) /\ O e. B ) -> ( ( Id ` C ) ` O ) e. ( O H O ) )
15		fvex 6201	. . . . . . . . . . . . 13 |- ( ( Id ` C ) ` O ) e. _V
16	15	elsn 4192	. . . . . . . . . . . 12 |- ( ( ( Id ` C ) ` O ) e. { h } <-> ( ( Id ` C ) ` O ) = h )
17		eqcom 2629	. . . . . . . . . . . 12 |- ( ( ( Id ` C ) ` O ) = h <-> h = ( ( Id ` C ) ` O ) )
18		vex 3203	. . . . . . . . . . . . 13 |- h e. _V
19		sneqbg 4374	. . . . . . . . . . . . . 14 |- ( h e. _V -> ( { h } = { ( ( Id ` C ) ` O ) } <-> h = ( ( Id ` C ) ` O ) ) )
20	19	bicomd 213	. . . . . . . . . . . . 13 |- ( h e. _V -> ( h = ( ( Id ` C ) ` O ) <-> { h } = { ( ( Id ` C ) ` O ) } ) )
21	18, 20	ax-mp 5	. . . . . . . . . . . 12 |- ( h = ( ( Id ` C ) ` O ) <-> { h } = { ( ( Id ` C ) ` O ) } )
22	16, 17, 21	3bitri 286	. . . . . . . . . . 11 |- ( ( ( Id ` C ) ` O ) e. { h } <-> { h } = { ( ( Id ` C ) ` O ) } )
23	22	biimpi 206	. . . . . . . . . 10 |- ( ( ( Id ` C ) ` O ) e. { h } -> { h } = { ( ( Id ` C ) ` O ) } )
24	23	a1i 11	. . . . . . . . 9 |- ( ( O H O ) = { h } -> ( ( ( Id ` C ) ` O ) e. { h } -> { h } = { ( ( Id ` C ) ` O ) } ) )
25		eleq2 2690	. . . . . . . . 9 |- ( ( O H O ) = { h } -> ( ( ( Id ` C ) ` O ) e. ( O H O ) <-> ( ( Id ` C ) ` O ) e. { h } ) )
26		eqeq1 2626	. . . . . . . . 9 |- ( ( O H O ) = { h } -> ( ( O H O ) = { ( ( Id ` C ) ` O ) } <-> { h } = { ( ( Id ` C ) ` O ) } ) )
27	24, 25, 26	3imtr4d 283	. . . . . . . 8 |- ( ( O H O ) = { h } -> ( ( ( Id ` C ) ` O ) e. ( O H O ) -> ( O H O ) = { ( ( Id ` C ) ` O ) } ) )
28	14, 27	syl5 34	. . . . . . 7 |- ( ( O H O ) = { h } -> ( ( ( ph /\ O e. (InitO ` C ) ) /\ O e. B ) -> ( O H O ) = { ( ( Id ` C ) ` O ) } ) )
29	28	exlimiv 1858	. . . . . 6 |- ( E. h ( O H O ) = { h } -> ( ( ( ph /\ O e. (InitO ` C ) ) /\ O e. B ) -> ( O H O ) = { ( ( Id ` C ) ` O ) } ) )
30	29	com12 32	. . . . 5 |- ( ( ( ph /\ O e. (InitO ` C ) ) /\ O e. B ) -> ( E. h ( O H O ) = { h } -> ( O H O ) = { ( ( Id ` C ) ` O ) } ) )
31	10, 30	syl5bi 232	. . . 4 |- ( ( ( ph /\ O e. (InitO ` C ) ) /\ O e. B ) -> ( E! h h e. ( O H O ) -> ( O H O ) = { ( ( Id ` C ) ` O ) } ) )
32	9, 31	syld 47	. . 3 |- ( ( ( ph /\ O e. (InitO ` C ) ) /\ O e. B ) -> ( A. o e. B E! h h e. ( O H o ) -> ( O H O ) = { ( ( Id ` C ) ` O ) } ) )
33	32	expimpd 629	. 2 |- ( ( ph /\ O e. (InitO ` C ) ) -> ( ( O e. B /\ A. o e. B E! h h e. ( O H o ) ) -> ( O H O ) = { ( ( Id ` C ) ` O ) } ) )
34	4, 33	mpd 15	1 |- ( ( ph /\ O e. (InitO ` C ) ) -> ( O H O ) = { ( ( Id ` C ) ` O ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   A.wral 2912   _Vcvv 3200   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Catccat 16325   Idccid 16326  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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-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-riota 6611  df-ov 6653  df-cat 16329  df-cid 16330  df-inito 16641

This theorem is referenced by:  2initoinv  16660