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Theorem invfun 17715
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 𝐡 = (Baseβ€˜πΆ)
invfval.n 𝑁 = (Invβ€˜πΆ)
invfval.c (πœ‘ β†’ 𝐢 ∈ Cat)
invfval.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
invfval.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
Assertion
Ref Expression
invfun (πœ‘ β†’ Fun (π‘‹π‘π‘Œ))
Proof of Theorem invfun
Dummy variables 𝑓 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfval.b . . . 4 𝐡 = (Baseβ€˜πΆ)
2 invfval.n . . . 4 𝑁 = (Invβ€˜πΆ)
3 invfval.c . . . 4 (πœ‘ β†’ 𝐢 ∈ Cat)
4 invfval.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐡)
5 invfval.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝐡)
6 eqid 2730 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
71, 2, 3, 4, 5, 6invss 17712 . . 3 (πœ‘ β†’ (π‘‹π‘π‘Œ) βŠ† ((𝑋(Hom β€˜πΆ)π‘Œ) Γ— (π‘Œ(Hom β€˜πΆ)𝑋)))
8 relxp 5693 . . 3 Rel ((𝑋(Hom β€˜πΆ)π‘Œ) Γ— (π‘Œ(Hom β€˜πΆ)𝑋))
9 relss 5780 . . 3 ((π‘‹π‘π‘Œ) βŠ† ((𝑋(Hom β€˜πΆ)π‘Œ) Γ— (π‘Œ(Hom β€˜πΆ)𝑋)) β†’ (Rel ((𝑋(Hom β€˜πΆ)π‘Œ) Γ— (π‘Œ(Hom β€˜πΆ)𝑋)) β†’ Rel (π‘‹π‘π‘Œ)))
107, 8, 9mpisyl 21 . 2 (πœ‘ β†’ Rel (π‘‹π‘π‘Œ))
11 eqid 2730 . . . . . 6 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
123adantr 479 . . . . . 6 ((πœ‘ ∧ (𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž)) β†’ 𝐢 ∈ Cat)
135adantr 479 . . . . . 6 ((πœ‘ ∧ (𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž)) β†’ π‘Œ ∈ 𝐡)
144adantr 479 . . . . . 6 ((πœ‘ ∧ (𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž)) β†’ 𝑋 ∈ 𝐡)
151, 2, 3, 4, 5, 11isinv 17711 . . . . . . . 8 (πœ‘ β†’ (𝑓(π‘‹π‘π‘Œ)𝑔 ↔ (𝑓(𝑋(Sectβ€˜πΆ)π‘Œ)𝑔 ∧ 𝑔(π‘Œ(Sectβ€˜πΆ)𝑋)𝑓)))
1615simplbda 498 . . . . . . 7 ((πœ‘ ∧ 𝑓(π‘‹π‘π‘Œ)𝑔) β†’ 𝑔(π‘Œ(Sectβ€˜πΆ)𝑋)𝑓)
1716adantrr 713 . . . . . 6 ((πœ‘ ∧ (𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž)) β†’ 𝑔(π‘Œ(Sectβ€˜πΆ)𝑋)𝑓)
181, 2, 3, 4, 5, 11isinv 17711 . . . . . . . 8 (πœ‘ β†’ (𝑓(π‘‹π‘π‘Œ)β„Ž ↔ (𝑓(𝑋(Sectβ€˜πΆ)π‘Œ)β„Ž ∧ β„Ž(π‘Œ(Sectβ€˜πΆ)𝑋)𝑓)))
1918simprbda 497 . . . . . . 7 ((πœ‘ ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž) β†’ 𝑓(𝑋(Sectβ€˜πΆ)π‘Œ)β„Ž)
2019adantrl 712 . . . . . 6 ((πœ‘ ∧ (𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž)) β†’ 𝑓(𝑋(Sectβ€˜πΆ)π‘Œ)β„Ž)
211, 11, 12, 13, 14, 17, 20sectcan 17706 . . . . 5 ((πœ‘ ∧ (𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž)) β†’ 𝑔 = β„Ž)
2221ex 411 . . . 4 (πœ‘ β†’ ((𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
2322alrimiv 1928 . . 3 (πœ‘ β†’ βˆ€β„Ž((𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
2423alrimivv 1929 . 2 (πœ‘ β†’ βˆ€π‘“βˆ€π‘”βˆ€β„Ž((𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
25 dffun2 6552 . 2 (Fun (π‘‹π‘π‘Œ) ↔ (Rel (π‘‹π‘π‘Œ) ∧ βˆ€π‘“βˆ€π‘”βˆ€β„Ž((𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž) β†’ 𝑔 = β„Ž)))
2610, 24, 25sylanbrc 581 1 (πœ‘ β†’ Fun (π‘‹π‘π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394  βˆ€wal 1537   = wceq 1539   ∈ wcel 2104   βŠ† wss 3947   class class class wbr 5147   Γ— cxp 5673  Rel wrel 5680  Fun wfun 6536  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  Hom chom 17212  Catccat 17612  Sectcsect 17695  Invcinv 17696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-cat 17616  df-cid 17617  df-sect 17698  df-inv 17699
This theorem is referenced by:  inviso1  17717  invf  17719  invco  17722  idinv  17740  ffthiso  17884  fuciso  17932  setciso  18045  catciso  18065  rngciso  46968  rngcisoALTV  46980  ringciso  47019  ringcisoALTV  47043
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