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Theorem invfun 17026
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 2798 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
71, 2, 3, 4, 5, 6invss 17023 . . 3 (𝜑 → (𝑋𝑁𝑌) ⊆ ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋)))
8 relxp 5537 . . 3 Rel ((𝑋(Hom ‘𝐶)𝑌) × (𝑌(Hom ‘𝐶)𝑋))
9 relss 5620 . . 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 2798 . . . . . 6 (Sect‘𝐶) = (Sect‘𝐶)
123adantr 484 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝐶 ∈ Cat)
135adantr 484 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑌𝐵)
144adantr 484 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑋𝐵)
151, 2, 3, 4, 5, 11isinv 17022 . . . . . . . 8 (𝜑 → (𝑓(𝑋𝑁𝑌)𝑔 ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)𝑔𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)))
1615simplbda 503 . . . . . . 7 ((𝜑𝑓(𝑋𝑁𝑌)𝑔) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
1716adantrr 716 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑔(𝑌(Sect‘𝐶)𝑋)𝑓)
181, 2, 3, 4, 5, 11isinv 17022 . . . . . . . 8 (𝜑 → (𝑓(𝑋𝑁𝑌) ↔ (𝑓(𝑋(Sect‘𝐶)𝑌)(𝑌(Sect‘𝐶)𝑋)𝑓)))
1918simprbda 502 . . . . . . 7 ((𝜑𝑓(𝑋𝑁𝑌)) → 𝑓(𝑋(Sect‘𝐶)𝑌))
2019adantrl 715 . . . . . 6 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑓(𝑋(Sect‘𝐶)𝑌))
211, 11, 12, 13, 14, 17, 20sectcan 17017 . . . . 5 ((𝜑 ∧ (𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌))) → 𝑔 = )
2221ex 416 . . . 4 (𝜑 → ((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = ))
2322alrimiv 1928 . . 3 (𝜑 → ∀((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = ))
2423alrimivv 1929 . 2 (𝜑 → ∀𝑓𝑔((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = ))
25 dffun2 6334 . 2 (Fun (𝑋𝑁𝑌) ↔ (Rel (𝑋𝑁𝑌) ∧ ∀𝑓𝑔((𝑓(𝑋𝑁𝑌)𝑔𝑓(𝑋𝑁𝑌)) → 𝑔 = )))
2610, 24, 25sylanbrc 586 1 (𝜑 → Fun (𝑋𝑁𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536   = wceq 1538  wcel 2111  wss 3881   class class class wbr 5030   × cxp 5517  Rel wrel 5524  Fun wfun 6318  cfv 6324  (class class class)co 7135  Basecbs 16475  Hom chom 16568  Catccat 16927  Sectcsect 17006  Invcinv 17007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-cat 16931  df-cid 16932  df-sect 17009  df-inv 17010
This theorem is referenced by:  inviso1  17028  invf  17030  invco  17033  idinv  17051  ffthiso  17191  fuciso  17237  setciso  17343  catciso  17359  rngciso  44604  rngcisoALTV  44616  ringciso  44655  ringcisoALTV  44679
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