Theorem invfun 17573

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)
invfval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
invfval.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		invfval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		invfval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		eqid 2736	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7	1, 2, 3, 4, 5, 6	invss 17570	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)))
8		relxp 5638	. . 3 ⊢ Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋))
9		relss 5723	. . 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 2736	. . . . . 6 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
12	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝐶 ∈ Cat)
13	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑌 ∈ 𝐵)
14	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑋 ∈ 𝐵)
15	1, 2, 3, 4, 5, 11	isinv 17569	. . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)𝑔 ∧ 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)))
16	15	simplbda 500	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)𝑔) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
17	16	adantrr 714	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
18	1, 2, 3, 4, 5, 11	isinv 17569	. . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)ℎ ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)ℎ ∧ ℎ(𝑌(Sect‘𝐶)𝑋)𝑓)))
19	18	simprbda 499	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ)
20	19	adantrl 713	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ)
21	1, 11, 12, 13, 14, 17, 20	sectcan 17564	. . . . 5 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔 = ℎ)
22	21	ex 413	. . . 4 ⊢ (𝜑 → ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
23	22	alrimiv 1929	. . 3 ⊢ (𝜑 → ∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
24	23	alrimivv 1930	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
25		dffun2 6489	. 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 396  ∀wal 1538   = wceq 1540   ∈ wcel 2105   ⊆ wss 3898   class class class wbr 5092   × cxp 5618  Rel wrel 5625  Fun wfun 6473  ‘cfv 6479  (class class class)co 7337  Basecbs 17009  Hom chom 17070  Catccat 17470  Sectcsect 17553  Invcinv 17554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-cat 17474  df-cid 17475  df-sect 17556  df-inv 17557

This theorem is referenced by:  inviso1  17575  invf  17577  invco  17580  idinv  17598  ffthiso  17742  fuciso  17790  setciso  17903  catciso  17923  rngciso  45891  rngcisoALTV  45903  ringciso  45942  ringcisoALTV  45966