Theorem inveq 17831

Description: If there are two inverses of a morphism, these inverses are equal. Corollary 3.11 of [Adamek] p. 28. (Contributed by AV, 10-Apr-2020.) (Revised by AV, 3-Jul-2022.)

Hypotheses

Ref	Expression
inveq.b	⊢ 𝐵 = (Base‘𝐶)
inveq.n	⊢ 𝑁 = (Inv‘𝐶)
inveq.c	⊢ (𝜑 → 𝐶 ∈ Cat)
inveq.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
inveq.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
inveq	⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))

Proof of Theorem inveq

Step	Hyp	Ref	Expression
1		inveq.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2769	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
3		inveq.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4	3	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐶 ∈ Cat)
5		inveq.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	5	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝑌 ∈ 𝐵)
7		inveq.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	7	adantr 485	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝑋 ∈ 𝐵)
9		inveq.n	. . . . . . . 8 ⊢ 𝑁 = (Inv‘𝐶)
10	1, 9, 3, 7, 5, 2	isinv 17817	. . . . . . 7 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		simpr 489	. . . . . . 7 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
12	10, 11	biimtrdi 256	. . . . . 6 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
13	12	com12 33	. . . . 5 ⊢ (𝐹(𝑋𝑁𝑌)𝐺 → (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
14	13	adantr 485	. . . 4 ⊢ ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
15	14	impcom 412	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
16	1, 9, 3, 7, 5, 2	isinv 17817	. . . . . 6 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐾 ∧ 𝐾(𝑌(Sect‘𝐶)𝑋)𝐹)))
17		simpl 487	. . . . . 6 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐾 ∧ 𝐾(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
18	16, 17	biimtrdi 256	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
19	18	adantld 495	. . . 4 ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
20	19	imp 411	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
21	1, 2, 4, 6, 8, 15, 20	sectcan 17812	. 2 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺 = 𝐾)
22	21	ex 417	1 ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  Catccat 17720  Sectcsect 17801  Invcinv 17802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-cat 17724  df-cid 17725  df-sect 17804  df-inv 17805

This theorem is referenced by: (None)