Theorem invfun 16247

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 2610	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
7	1, 2, 3, 4, 5, 6	invss 16244	. . 3 ⊢ (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)))
8		relxp 5150	. . 3 ⊢ Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋))
9		relss 5129	. . 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 2610	. . . . . 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 16243	. . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)𝑔 ∧ 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)))
16	15	simplbda 652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)𝑔) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
17	16	adantrr 749	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
18	1, 2, 3, 4, 5, 11	isinv 16243	. . . . . . . 8 ⊢ (𝜑 → (𝑓(𝑋𝑁𝑌)ℎ ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)ℎ ∧ ℎ(𝑌(Sect‘𝐶)𝑋)𝑓)))
19	18	simprbda 651	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ)
20	19	adantrl 748	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑓(𝑋(Sect‘𝐶)𝑌)ℎ)
21	1, 11, 12, 13, 14, 17, 20	sectcan 16238	. . . . 5 ⊢ ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ)) → 𝑔 = ℎ)
22	21	ex 449	. . . 4 ⊢ (𝜑 → ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
23	22	alrimiv 1842	. . 3 ⊢ (𝜑 → ∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
24	23	alrimivv 1843	. 2 ⊢ (𝜑 → ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ))
25		dffun2 5814	. 2 ⊢ (Fun (𝑋𝑁𝑌) ↔ (Rel (𝑋𝑁𝑌) ∧ ∀𝑓∀𝑔∀ℎ((𝑓(𝑋𝑁𝑌)𝑔 ∧ 𝑓(𝑋𝑁𝑌)ℎ) → 𝑔 = ℎ)))
26	10, 24, 25	sylanbrc 695	1 ⊢ (𝜑 → Fun (𝑋𝑁𝑌))

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   class class class wbr 4583   × cxp 5036  Rel wrel 5043  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Hom chom 15779  Catccat 16148  Sectcsect 16227  Invcinv 16228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-cat 16152  df-cid 16153  df-sect 16230  df-inv 16231

This theorem is referenced by:  inviso1  16249  invf  16251  invco  16254  idinv  16272  funciso  16357  ffthiso  16412  fuciso  16458  setciso  16564  catciso  16580  rngciso  41774  rngcisoALTV  41786  ringciso  41825  ringcisoALTV  41849