Theorem funcinv 17968

Description: The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
funcinv.b	⊢ 𝐵 = (Base‘𝐷)
funcinv.s	⊢ 𝐼 = (Inv‘𝐷)
funcinv.t	⊢ 𝐽 = (Inv‘𝐸)
funcinv.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
funcinv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
funcinv.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
funcinv.m	⊢ (𝜑 → 𝑀(𝑋𝐼𝑌)𝑁)

Assertion

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

Proof of Theorem funcinv

Step	Hyp	Ref	Expression
1		funcinv.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
2		eqid 2762	. . 3 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
3		eqid 2762	. . 3 ⊢ (Sect‘𝐸) = (Sect‘𝐸)
4		funcinv.f	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
5		funcinv.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		funcinv.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		funcinv.m	. . . . 5 ⊢ (𝜑 → 𝑀(𝑋𝐼𝑌)𝑁)
8		funcinv.s	. . . . . 6 ⊢ 𝐼 = (Inv‘𝐷)
9		df-br 5108	. . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
10	4, 9	sylib 221	. . . . . . . 8 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11		funcrcl 17958	. . . . . . . 8 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
12	10, 11	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
13	12	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
14	1, 8, 13, 5, 6, 2	isinv 17855	. . . . 5 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)))
15	7, 14	mpbid 235	. . . 4 ⊢ (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))
16	15	simpld 500	. . 3 ⊢ (𝜑 → 𝑀(𝑋(Sect‘𝐷)𝑌)𝑁)
17	1, 2, 3, 4, 5, 6, 16	funcsect 17967	. 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))
18	15	simprd 501	. . 3 ⊢ (𝜑 → 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)
19	1, 2, 3, 4, 6, 5, 18	funcsect 17967	. 2 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀))
20		eqid 2762	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
21		funcinv.t	. . 3 ⊢ 𝐽 = (Inv‘𝐸)
22	12	simprd 501	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
23	1, 20, 4	funcf1 17961	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸))
24	23, 5	ffvelcdmd 7082	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸))
25	23, 6	ffvelcdmd 7082	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸))
26	20, 21, 22, 24, 25, 3	isinv 17855	. 2 ⊢ (𝜑 → (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ↔ (((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁) ∧ ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀))))
27	17, 19, 26	mpbir2and 726	1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4593   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  Catccat 17758  Sectcsect 17839  Invcinv 17840   Func cfunc 17949

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-map 8832  df-ixp 8909  df-sect 17842  df-inv 17843  df-func 17953

This theorem is used by:  funciso  17969