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Theorem invfun 16345
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 (𝜑𝐶 ∈ Cat)
invfval.x (𝜑𝑋𝐵)
invfval.y (𝜑𝑌𝐵)
Assertion
Ref Expression
invfun (𝜑 → Fun (𝑋𝑁𝑌))

Proof of Theorem invfun
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfval.b . . . 4 𝐵 = (Base‘𝐶)
2 invfval.n . . . 4 𝑁 = (Inv‘𝐶)
3 invfval.c . . . 4 (𝜑𝐶 ∈ Cat)
4 invfval.x . . . 4 (𝜑𝑋𝐵)
5 invfval.y . . . 4 (𝜑𝑌𝐵)
6 eqid 2621 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
71, 2, 3, 4, 5, 6invss 16342 . . 3 (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)))
8 relxp 5188 . . 3 Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋))
9 relss 5167 . . 3 ((𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)) → (Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)) → Rel (𝑋𝑁𝑌)))
107, 8, 9mpisyl 21 . 2 (𝜑 → Rel (𝑋𝑁𝑌))
11 eqid 2621 . . . . . 6 (Sect‘𝐶) = (Sect‘𝐶)
123adantr 481 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝐶 ∈ Cat)
135adantr 481 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑌𝐵)
144adantr 481 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑋𝐵)
151, 2, 3, 4, 5, 11isinv 16341 . . . . . . . 8 (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)𝑔𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)))
1615simplbda 653 . . . . . . 7 ((𝜑𝑓(𝑋𝑁𝑌)𝑔) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
1716adantrr 752 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
181, 2, 3, 4, 5, 11isinv 16341 . . . . . . . 8 (𝜑 → (𝑓(𝑋𝑁𝑌) ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)(𝑌(Sect‘𝐶)𝑋)𝑓)))
1918simprbda 652 . . . . . . 7 ((𝜑𝑓(𝑋𝑁𝑌)) → 𝑓(𝑋(Sect‘𝐶)𝑌))
2019adantrl 751 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑓(𝑋(Sect‘𝐶)𝑌))
211, 11, 12, 13, 14, 17, 20sectcan 16336 . . . . 5 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑔 = )
2221ex 450 . . . 4 (𝜑 → ((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = ))
2322alrimiv 1852 . . 3 (𝜑 → ∀((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = ))
2423alrimivv 1853 . 2 (𝜑 → ∀𝑓𝑔((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = ))
25 dffun2 5857 . 2 (Fun (𝑋𝑁𝑌) ↔ (Rel (𝑋𝑁𝑌) ∧ ∀𝑓𝑔((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = )))
2610, 24, 25sylanbrc 697 1 (𝜑 → Fun (𝑋𝑁𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478   = wceq 1480  wcel 1987  wss 3555   class class class wbr 4613   × cxp 5072  Rel wrel 5079  Fun wfun 5841  cfv 5847  (class class class)co 6604  Basecbs 15781  Hom chom 15873  Catccat 16246  Sectcsect 16325  Invcinv 16326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-cat 16250  df-cid 16251  df-sect 16328  df-inv 16329
This theorem is referenced by:  inviso1  16347  invf  16349  invco  16352  idinv  16370  funciso  16455  ffthiso  16510  fuciso  16556  setciso  16662  catciso  16678  rngciso  41267  rngcisoALTV  41279  ringciso  41318  ringcisoALTV  41342
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