Theorem istermoi 16654

Description: Implication of a class being a terminal object. (Contributed by AV, 18-Apr-2020.)

Hypotheses

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

Assertion

Ref	Expression
istermoi	|- ( ( ph /\ O e. (TermO ` C ) ) -> ( O e. B /\ A. b e. B E! h h e. ( b H O ) ) )

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

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

Proof of Theorem istermoi

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isinitoi.c	. . . . . 6 |- ( ph -> C e. Cat )
2		isinitoi.b	. . . . . 6 |- B = ( Base ` C )
3		isinitoi.h	. . . . . 6 |- H = ( Hom ` C )
4	1, 2, 3	termoval 16648	. . . . 5 |- ( ph -> (TermO ` C ) = { a e. B | A. b e. B E! h h e. ( b H a ) } )
5	4	eleq2d 2687	. . . 4 |- ( ph -> ( O e. (TermO ` C ) <-> O e. { a e. B | A. b e. B E! h h e. ( b H a ) } ) )
6		elrabi 3359	. . . 4 |- ( O e. { a e. B | A. b e. B E! h h e. ( b H a ) } -> O e. B )
7	5, 6	syl6bi 243	. . 3 |- ( ph -> ( O e. (TermO ` C ) -> O e. B ) )
8	7	imp 445	. 2 |- ( ( ph /\ O e. (TermO ` C ) ) -> O e. B )
9	1	adantr 481	. . . . 5 |- ( ( ph /\ O e. B ) -> C e. Cat )
10		simpr 477	. . . . 5 |- ( ( ph /\ O e. B ) -> O e. B )
11	2, 3, 9, 10	istermo 16651	. . . 4 |- ( ( ph /\ O e. B ) -> ( O e. (TermO ` C ) <-> A. b e. B E! h h e. ( b H O ) ) )
12	11	biimpd 219	. . 3 |- ( ( ph /\ O e. B ) -> ( O e. (TermO ` C ) -> A. b e. B E! h h e. ( b H O ) ) )
13	12	impancom 456	. 2 |- ( ( ph /\ O e. (TermO ` C ) ) -> ( O e. B -> A. b e. B E! h h e. ( b H O ) ) )
14	8, 13	jcai 559	1 |- ( ( ph /\ O e. (TermO ` C ) ) -> ( O e. B /\ A. b e. B E! h h e. ( b H O ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E!weu 2470   A.wral 2912   {crab 2916   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952   Catccat 16325  TermOctermo 16639

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-termo 16642

This theorem is referenced by:  termoid  16656  termoo  16658  termoeu1  16668