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Theorem invfun 17711
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 2733 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
71, 2, 3, 4, 5, 6invss 17708 . . 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 5695 . . 3 Rel ((𝑋(Hom β€˜πΆ)π‘Œ) Γ— (π‘Œ(Hom β€˜πΆ)𝑋))
9 relss 5782 . . 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 2733 . . . . . 6 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
123adantr 482 . . . . . 6 ((πœ‘ ∧ (𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž)) β†’ 𝐢 ∈ Cat)
135adantr 482 . . . . . 6 ((πœ‘ ∧ (𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž)) β†’ π‘Œ ∈ 𝐡)
144adantr 482 . . . . . 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 17707 . . . . . . . 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 501 . . . . . . 7 ((πœ‘ ∧ 𝑓(π‘‹π‘π‘Œ)𝑔) β†’ 𝑔(π‘Œ(Sectβ€˜πΆ)𝑋)𝑓)
1716adantrr 716 . . . . . 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 17707 . . . . . . . 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 500 . . . . . . 7 ((πœ‘ ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž) β†’ 𝑓(𝑋(Sectβ€˜πΆ)π‘Œ)β„Ž)
2019adantrl 715 . . . . . 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 17702 . . . . 5 ((πœ‘ ∧ (𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž)) β†’ 𝑔 = β„Ž)
2221ex 414 . . . 4 (πœ‘ β†’ ((𝑓(π‘‹π‘π‘Œ)𝑔 ∧ 𝑓(π‘‹π‘π‘Œ)β„Ž) β†’ 𝑔 = β„Ž))
2322alrimiv 1931 . . 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 1932 . 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 6554 . 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 584 1 (πœ‘ β†’ Fun (π‘‹π‘π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107   βŠ† wss 3949   class class class wbr 5149   Γ— cxp 5675  Rel wrel 5682  Fun wfun 6538  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Hom chom 17208  Catccat 17608  Sectcsect 17691  Invcinv 17692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-cat 17612  df-cid 17613  df-sect 17694  df-inv 17695
This theorem is referenced by:  inviso1  17713  invf  17715  invco  17718  idinv  17736  ffthiso  17880  fuciso  17928  setciso  18041  catciso  18061  rngciso  46880  rngcisoALTV  46892  ringciso  46931  ringcisoALTV  46955
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