Theorem funcinv 17809

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 2737	. . 3 ⊢ (Sect‘𝐷) = (Sect‘𝐷)
3		eqid 2737	. . 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 5101	. . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
10	4, 9	sylib 218	. . . . . . . 8 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11		funcrcl 17799	. . . . . . . 8 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
13	12	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ Cat)
14	1, 8, 13, 5, 6, 2	isinv 17696	. . . . 5 ⊢ (𝜑 → (𝑀(𝑋𝐼𝑌)𝑁 ↔ (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)))
15	7, 14	mpbid 232	. . . 4 ⊢ (𝜑 → (𝑀(𝑋(Sect‘𝐷)𝑌)𝑁 ∧ 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀))
16	15	simpld 494	. . 3 ⊢ (𝜑 → 𝑀(𝑋(Sect‘𝐷)𝑌)𝑁)
17	1, 2, 3, 4, 5, 6, 16	funcsect 17808	. 2 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)(Sect‘𝐸)(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))
18	15	simprd 495	. . 3 ⊢ (𝜑 → 𝑁(𝑌(Sect‘𝐷)𝑋)𝑀)
19	1, 2, 3, 4, 6, 5, 18	funcsect 17808	. 2 ⊢ (𝜑 → ((𝑌𝐺𝑋)‘𝑁)((𝐹‘𝑌)(Sect‘𝐸)(𝐹‘𝑋))((𝑋𝐺𝑌)‘𝑀))
20		eqid 2737	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
21		funcinv.t	. . 3 ⊢ 𝐽 = (Inv‘𝐸)
22	12	simprd 495	. . 3 ⊢ (𝜑 → 𝐸 ∈ Cat)
23	1, 20, 4	funcf1 17802	. . . 4 ⊢ (𝜑 → 𝐹:𝐵⟶(Base‘𝐸))
24	23, 5	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (Base‘𝐸))
25	23, 6	ffvelcdmd 7039	. . 3 ⊢ (𝜑 → (𝐹‘𝑌) ∈ (Base‘𝐸))
26	20, 21, 22, 24, 25, 3	isinv 17696	. 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 714	1 ⊢ (𝜑 → ((𝑋𝐺𝑌)‘𝑀)((𝐹‘𝑋)𝐽(𝐹‘𝑌))((𝑌𝐺𝑋)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Catccat 17599  Sectcsect 17680  Invcinv 17681   Func cfunc 17790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777  df-ixp 8848  df-sect 17683  df-inv 17684  df-func 17794

This theorem is referenced by:  funciso  17810