Theorem inveq 17808

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 2763	. . 3 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
3		inveq.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4	3	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐶 ∈ Cat)
5		inveq.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6	5	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝑌 ∈ 𝐵)
7		inveq.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8	7	adantr 484	. . 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 17794	. . . . . . 7 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)))
11		simpr 488	. . . . . . 7 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐺 ∧ 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
12	10, 11	biimtrdi 255	. . . . . 6 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
13	12	com12 32	. . . . 5 ⊢ (𝐹(𝑋𝑁𝑌)𝐺 → (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
14	13	adantr 484	. . . 4 ⊢ ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → (𝜑 → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹))
15	14	impcom 411	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺(𝑌(Sect‘𝐶)𝑋)𝐹)
16	1, 9, 3, 7, 5, 2	isinv 17794	. . . . . 6 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 ↔ (𝐹(𝑋(Sect‘𝐶)𝑌)𝐾 ∧ 𝐾(𝑌(Sect‘𝐶)𝑋)𝐹)))
17		simpl 486	. . . . . 6 ⊢ ((𝐹(𝑋(Sect‘𝐶)𝑌)𝐾 ∧ 𝐾(𝑌(Sect‘𝐶)𝑋)𝐹) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
18	16, 17	biimtrdi 255	. . . . 5 ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐾 → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
19	18	adantld 494	. . . 4 ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾))
20	19	imp 410	. . 3 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐹(𝑋(Sect‘𝐶)𝑌)𝐾)
21	1, 2, 4, 6, 8, 15, 20	sectcan 17789	. 2 ⊢ ((𝜑 ∧ (𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾)) → 𝐺 = 𝐾)
22	21	ex 416	1 ⊢ (𝜑 → ((𝐹(𝑋𝑁𝑌)𝐺 ∧ 𝐹(𝑋𝑁𝑌)𝐾) → 𝐺 = 𝐾))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143   class class class wbr 5101  ‘cfv 6522  (class class class)co 7397  Basecbs 17246  Catccat 17697  Sectcsect 17778  Invcinv 17779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-1st 7971  df-2nd 7972  df-cat 17701  df-cid 17702  df-sect 17781  df-inv 17782

This theorem is referenced by: (None)