Theorem fthinv 18010

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 2766	. . . 4 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
9		eqid 2766	. . . 4 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	fthsect 18009	. . 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 18009	. . 3 ⊢ (𝜑 → (𝑁(𝑌(Sect‘𝐶)𝑋)𝑀 ↔ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐷)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀)))
12	10, 11	anbi12d 644	. 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 17991	. . . . . . . 8 ⊢ (𝐶 Faith 𝐷) ⊆ (𝐶 Func 𝐷)
15	14	ssbri 5161	. . . . . . 7 ⊢ (𝐹(𝐶 Faith 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
16	3, 15	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
17		df-br 5115	. . . . . 6 ⊢ (𝐹(𝐶 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
18	16, 17	sylib 221	. . . . 5 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷))
19		funcrcl 17945	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
20	18, 19	syl 18	. . . 4 ⊢ (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
21	20	simpld 500	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
22	1, 13, 21, 4, 5, 8	isinv 17842	. 2 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐶)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐶)𝑋)𝑀)))
23		eqid 2766	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
24		fthinv.t	. . 3 ⊢ 𝐽 = (Inv‘𝐷)
25	20	simprd 501	. . 3 ⊢ (𝜑 → 𝐷 ∈ Cat)
26	1, 23, 16	funcf1 17948	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐷))
27	26, 4	ffvelcdmd 7087	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐷))
28	26, 5	ffvelcdmd 7087	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐷))
29	23, 24, 25, 27, 28, 9	isinv 17842	. 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 314	1 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ⟨cop 4600   class class class wbr 5114  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  Hom chom 17346  Catccat 17745  Sectcsect 17826  Invcinv 17827   Func cfunc 17936   Faith cfth 17987

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-map 8835  df-ixp 8905  df-cat 17749  df-cid 17750  df-sect 17829  df-inv 17830  df-func 17940  df-fth 17989

This theorem is used by:  ffthiso  18013