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Theorem invfun 17780
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 2726 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
71, 2, 3, 4, 5, 6invss 17777 . . 3 (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)))
8 relxp 5700 . . 3 Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋))
9 relss 5787 . . 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 2726 . . . . . 6 (Sect‘𝐶) = (Sect‘𝐶)
123adantr 479 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝐶 ∈ Cat)
135adantr 479 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑌𝐵)
144adantr 479 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑋𝐵)
151, 2, 3, 4, 5, 11isinv 17776 . . . . . . . 8 (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)𝑔𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)))
1615simplbda 498 . . . . . . 7 ((𝜑𝑓(𝑋𝑁𝑌)𝑔) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
1716adantrr 715 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
181, 2, 3, 4, 5, 11isinv 17776 . . . . . . . 8 (𝜑 → (𝑓(𝑋𝑁𝑌) ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)(𝑌(Sect‘𝐶)𝑋)𝑓)))
1918simprbda 497 . . . . . . 7 ((𝜑𝑓(𝑋𝑁𝑌)) → 𝑓(𝑋(Sect‘𝐶)𝑌))
2019adantrl 714 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑓(𝑋(Sect‘𝐶)𝑌))
211, 11, 12, 13, 14, 17, 20sectcan 17771 . . . . 5 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑔 = )
2221ex 411 . . . 4 (𝜑 → ((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = ))
2322alrimiv 1923 . . 3 (𝜑 → ∀((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = ))
2423alrimivv 1924 . 2 (𝜑 → ∀𝑓𝑔((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = ))
25 dffun2 6564 . 2 (Fun (𝑋𝑁𝑌) ↔ (Rel (𝑋𝑁𝑌) ∧ ∀𝑓𝑔((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = )))
2610, 24, 25sylanbrc 581 1 (𝜑 → Fun (𝑋𝑁𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1532   = wceq 1534  wcel 2099  wss 3947   class class class wbr 5153   × cxp 5680  Rel wrel 5687  Fun wfun 6548  cfv 6554  (class class class)co 7424  Basecbs 17213  Hom chom 17277  Catccat 17677  Sectcsect 17760  Invcinv 17761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-cat 17681  df-cid 17682  df-sect 17763  df-inv 17764
This theorem is referenced by:  inviso1  17782  invf  17784  invco  17787  idinv  17805  ffthiso  17951  fuciso  18000  setciso  18113  catciso  18133  rngciso  20616  ringciso  20650  rngcisoALTV  47654  ringcisoALTV  47688
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