Theorem fthinv 17029

Description: A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)

Hypotheses

Ref	Expression
fthsect.b	⊢ 𝐵 = (Base‘𝐶)
fthsect.h	⊢ 𝐻 = (Hom ‘𝐶)
fthsect.f	⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
fthsect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
fthsect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
fthsect.m	⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
fthsect.n	⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑋))
fthinv.s	⊢ 𝐼 = (Inv‘𝐶)
fthinv.t	⊢ 𝐽 = (Inv‘𝐷)

Assertion

Ref	Expression
fthinv	⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)))

Proof of Theorem fthinv

Step	Hyp	Ref	Expression
1		fthsect.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		fthsect.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
3		fthsect.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Faith 𝐷)𝐺)
4		fthsect.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		fthsect.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		fthsect.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (𝑋𝐻𝑌))
7		fthsect.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝑌𝐻𝑋))
8		eqid 2797	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
9		eqid 2797	. . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	fthsect 17028	. . 3 ⊢ (𝜑 → (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐷)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)))
11	1, 2, 3, 5, 4, 7, 6, 8, 9	fthsect 17028	. . 3 ⊢ (𝜑 → (𝑁(𝑌(Sect‘𝐶)𝑋)𝑀 ↔ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))
12	10, 11	anbi12d 630	. 2 ⊢ (𝜑 → ((𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐶)𝑋)𝑀) ↔ (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐷)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ∧ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀))))
13		fthinv.s	. . 3 ⊢ 𝐼 = (Inv‘𝐶)
14		fthfunc 17010	. . . . . . . 8 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
15	14	ssbri 5013	. . . . . . 7 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
16	3, 15	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
17		df-br 4969	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
18	16, 17	sylib 219	. . . . 5 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
19		funcrcl 16966	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
20	18, 19	syl 17	. . . 4 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
21	20	simpld 495	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
22	1, 13, 21, 4, 5, 8	isinv 16863	. 2 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐶)𝑋)𝑀)))
23		eqid 2797	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
24		fthinv.t	. . 3 ⊢ 𝐽 = (Inv‘𝐷)
25	20	simprd 496	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
26	1, 23, 16	funcf1 16969	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷))
27	26, 4	ffvelrnd 6724	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
28	26, 5	ffvelrnd 6724	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
29	23, 24, 25, 27, 28, 9	isinv 16863	. 2 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐷)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ∧ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀))))
30	12, 22, 29	3bitr4d 312	1 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083  ⟨cop 4484   class class class wbr 4968  ‘cfv 6232  (class class class)co 7023  Basecbs 16316  Hom chom 16409  Catccat 16768  Sectcsect 16847  Invcinv 16848   Func cfunc 16957   Faith cfth 17006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-map 8265  df-ixp 8318  df-cat 16772  df-cid 16773  df-sect 16850  df-inv 16851  df-func 16961  df-fth 17008

This theorem is referenced by:  ffthiso  17032