Theorem invfun 17777

Description: The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
invfun	⊢ (𝜑 → Fun (𝑋𝑁𝑌))

Proof of Theorem invfun

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		invfval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		invfval.n	. . . 4 ⊢ 𝑁 = (Inv‘𝐶)
3		invfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
4		invss.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		invss.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		eqid 2735	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7	1, 2, 3, 4, 5, 6	invss 17774	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)))
8		relxp 5672	. . 3 ⊢ Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋))
9		relss 5760	. . 3 ⊢ ((𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)) → (Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)) → Rel (𝑋𝑁𝑌)))
10	7, 8, 9	mpisyl 21	. 2 ⊢ (𝜑 → Rel (𝑋𝑁𝑌))
11		eqid 2735	. . . . . 6 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
12	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝐶 ∈ Cat)
13	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑌 ∈ 𝐵)
14	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑋 ∈ 𝐵)
15	1, 2, 3, 4, 5, 11	isinv 17773	. . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)𝑔 ∧ 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)))
16	15	simplbda 499	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)𝑔) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
17	16	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
18	1, 2, 3, 4, 5, 11	isinv 17773	. . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)ℎ ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)ℎ ∧ ℎ(𝑌(Sect‘𝐶)𝑋)𝑓)))
19	18	simprbda 498	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ)
20	19	adantrl 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ)
21	1, 11, 12, 13, 14, 17, 20	sectcan 17768	. . . . 5 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔 = ℎ)
22	21	ex 412	. . . 4 ⊢ (𝜑 → ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
23	22	alrimiv 1927	. . 3 ⊢ (𝜑 → ∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
24	23	alrimivv 1928	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
25		dffun2 6541	. 2 ⊢ (Fun (𝑋𝑁𝑌) ↔ (Rel (𝑋𝑁𝑌) ∧ ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ)))
26	10, 24, 25	sylanbrc 583	1 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108   ⊆ wss 3926   class class class wbr 5119   × cxp 5652  Rel wrel 5659  Fun wfun 6525  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  Hom chom 17282  Catccat 17676  Sectcsect 17757  Invcinv 17758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-cat 17680  df-cid 17681  df-sect 17760  df-inv 17761

This theorem is referenced by:  inviso1  17779  invf  17781  invco  17784  idinv  17802  ffthiso  17944  fuciso  17991  setciso  18104  catciso  18124  rngciso  20598  ringciso  20632  rngcisoALTV  48252  ringcisoALTV  48286