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Theorem 2termoinv 16667
Description: Morphisms between two terminal objects are inverses. (Contributed by AV, 18-Apr-2020.)
Hypotheses
Ref Expression
termoeu1.c |- ( ph -> C e. Cat )
termoeu1.a |- ( ph -> A e. (TermO ` C ) )
termoeu1.b |- ( ph -> B e. (TermO ` C ) )
Assertion
Ref Expression
2termoinv |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> F ( A (Inv ` C ) B ) G )
Proof of Theorem 2termoinv
StepHypRef Expression
1 eqid 2622 . . . . 5 |- ( Base ` C ) = ( Base ` C )
2 eqid 2622 . . . . 5 |- ( Hom ` C ) = ( Hom ` C )
3 eqid 2622 . . . . 5 |- (comp ` C ) = (comp ` C )
4 termoeu1.c . . . . . 6 |- ( ph -> C e. Cat )
543ad2ant1 1082 . . . . 5 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> C e. Cat )
6 termoeu1.a . . . . . . 7 |- ( ph -> A e. (TermO ` C ) )
7 termoo 16658 . . . . . . 7 |- ( C e. Cat -> ( A e. (TermO ` C ) -> A e. ( Base ` C ) ) )
84, 6, 7sylc 65 . . . . . 6 |- ( ph -> A e. ( Base ` C ) )
983ad2ant1 1082 . . . . 5 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> A e. ( Base ` C ) )
10 termoeu1.b . . . . . . 7 |- ( ph -> B e. (TermO ` C ) )
11 termoo 16658 . . . . . . 7 |- ( C e. Cat -> ( B e. (TermO ` C ) -> B e. ( Base ` C ) ) )
124, 10, 11sylc 65 . . . . . 6 |- ( ph -> B e. ( Base ` C ) )
13123ad2ant1 1082 . . . . 5 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> B e. ( Base ` C ) )
14 simp3 1063 . . . . 5 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> F e. ( A ( Hom ` C ) B ) )
15 simp2 1062 . . . . 5 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> G e. ( B ( Hom ` C ) A ) )
161, 2, 3, 5, 9, 13, 9, 14, 15catcocl 16346 . . . 4 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( G ( <. A , B >. (comp ` C ) A ) F ) e. ( A ( Hom ` C ) A ) )
171, 2, 4termoid 16656 . . . . . . . 8 |- ( ( ph /\ A e. (TermO ` C ) ) -> ( A ( Hom ` C ) A ) = { ( ( Id ` C ) ` A ) } )
186, 17mpdan 702 . . . . . . 7 |- ( ph -> ( A ( Hom ` C ) A ) = { ( ( Id ` C ) ` A ) } )
19183ad2ant1 1082 . . . . . 6 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( A ( Hom ` C ) A ) = { ( ( Id ` C ) ` A ) } )
2019eleq2d 2687 . . . . 5 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( ( G ( <. A , B >. (comp ` C ) A ) F ) e. ( A ( Hom ` C ) A ) <-> ( G ( <. A , B >. (comp ` C ) A ) F ) e. { ( ( Id ` C ) ` A ) } ) )
21 elsni 4194 . . . . 5 |- ( ( G ( <. A , B >. (comp ` C ) A ) F ) e. { ( ( Id ` C ) ` A ) } -> ( G ( <. A , B >. (comp ` C ) A ) F ) = ( ( Id ` C ) ` A ) )
2220, 21syl6bi 243 . . . 4 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( ( G ( <. A , B >. (comp ` C ) A ) F ) e. ( A ( Hom ` C ) A ) -> ( G ( <. A , B >. (comp ` C ) A ) F ) = ( ( Id ` C ) ` A ) ) )
2316, 22mpd 15 . . 3 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( G ( <. A , B >. (comp ` C ) A ) F ) = ( ( Id ` C ) ` A ) )
24 eqid 2622 . . . 4 |- ( Id ` C ) = ( Id ` C )
25 eqid 2622 . . . 4 |- (Sect ` C ) = (Sect ` C )
261, 2, 3, 24, 25, 5, 9, 13, 14, 15issect2 16414 . . 3 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( F ( A (Sect ` C ) B ) G <-> ( G ( <. A , B >. (comp ` C ) A ) F ) = ( ( Id ` C ) ` A ) ) )
2723, 26mpbird 247 . 2 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> F ( A (Sect ` C ) B ) G )
281, 2, 3, 5, 13, 9, 13, 15, 14catcocl 16346 . . . 4 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( F ( <. B , A >. (comp ` C ) B ) G ) e. ( B ( Hom ` C ) B ) )
291, 2, 4termoid 16656 . . . . . . . 8 |- ( ( ph /\ B e. (TermO ` C ) ) -> ( B ( Hom ` C ) B ) = { ( ( Id ` C ) ` B ) } )
3010, 29mpdan 702 . . . . . . 7 |- ( ph -> ( B ( Hom ` C ) B ) = { ( ( Id ` C ) ` B ) } )
31303ad2ant1 1082 . . . . . 6 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( B ( Hom ` C ) B ) = { ( ( Id ` C ) ` B ) } )
3231eleq2d 2687 . . . . 5 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( ( F ( <. B , A >. (comp ` C ) B ) G ) e. ( B ( Hom ` C ) B ) <-> ( F ( <. B , A >. (comp ` C ) B ) G ) e. { ( ( Id ` C ) ` B ) } ) )
33 elsni 4194 . . . . 5 |- ( ( F ( <. B , A >. (comp ` C ) B ) G ) e. { ( ( Id ` C ) ` B ) } -> ( F ( <. B , A >. (comp ` C ) B ) G ) = ( ( Id ` C ) ` B ) )
3432, 33syl6bi 243 . . . 4 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( ( F ( <. B , A >. (comp ` C ) B ) G ) e. ( B ( Hom ` C ) B ) -> ( F ( <. B , A >. (comp ` C ) B ) G ) = ( ( Id ` C ) ` B ) ) )
3528, 34mpd 15 . . 3 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( F ( <. B , A >. (comp ` C ) B ) G ) = ( ( Id ` C ) ` B ) )
361, 2, 3, 24, 25, 5, 13, 9, 15, 14issect2 16414 . . 3 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( G ( B (Sect ` C ) A ) F <-> ( F ( <. B , A >. (comp ` C ) B ) G ) = ( ( Id ` C ) ` B ) ) )
3735, 36mpbird 247 . 2 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> G ( B (Sect ` C ) A ) F )
38 eqid 2622 . . . 4 |- (Inv ` C ) = (Inv ` C )
391, 38, 4, 8, 12, 25isinv 16420 . . 3 |- ( ph -> ( F ( A (Inv ` C ) B ) G <-> ( F ( A (Sect ` C ) B ) G /\ G ( B (Sect ` C ) A ) F ) ) )
40393ad2ant1 1082 . 2 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> ( F ( A (Inv ` C ) B ) G <-> ( F ( A (Sect ` C ) B ) G /\ G ( B (Sect ` C ) A ) F ) ) )
4127, 37, 40mpbir2and 957 1 |- ( ( ph /\ G e. ( B ( Hom ` C ) A ) /\ F e. ( A ( Hom ` C ) B ) ) -> F ( A (Inv ` C ) B ) G )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  Sectcsect 16404  Invcinv 16405  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-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-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-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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-cat 16329  df-cid 16330  df-sect 16407  df-inv 16408  df-termo 16642
This theorem is referenced by:  termoeu1  16668
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