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Theorem isoval 17813
Description: The isomorphisms are the domain of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof shortened by AV, 21-May-2020.)
Hypotheses
Ref Expression
invfval.b 𝐵 = (Base‘𝐶)
invfval.n 𝑁 = (Inv‘𝐶)
invfval.c (𝜑𝐶 ∈ Cat)
invfval.x (𝜑𝑋𝐵)
invfval.y (𝜑𝑌𝐵)
isoval.n 𝐼 = (Iso‘𝐶)
Assertion
Ref Expression
isoval (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))

Proof of Theorem isoval
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invfval.c . . . . 5 (𝜑𝐶 ∈ Cat)
2 isofval 17805 . . . . 5 (𝐶 ∈ Cat → (Iso‘𝐶) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶)))
31, 2syl 17 . . . 4 (𝜑 → (Iso‘𝐶) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶)))
4 isoval.n . . . 4 𝐼 = (Iso‘𝐶)
5 invfval.n . . . . 5 𝑁 = (Inv‘𝐶)
65coeq2i 5874 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶))
73, 4, 63eqtr4g 2800 . . 3 (𝜑𝐼 = ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁))
87oveqd 7448 . 2 (𝜑 → (𝑋𝐼𝑌) = (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌))
9 eqid 2735 . . . . . 6 (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)))
10 ovex 7464 . . . . . . 7 (𝑥(Sect‘𝐶)𝑦) ∈ V
1110inex1 5323 . . . . . 6 ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)) ∈ V
129, 11fnmpoi 8094 . . . . 5 (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) Fn (𝐵 × 𝐵)
13 invfval.b . . . . . . 7 𝐵 = (Base‘𝐶)
14 invfval.x . . . . . . 7 (𝜑𝑋𝐵)
15 invfval.y . . . . . . 7 (𝜑𝑌𝐵)
16 eqid 2735 . . . . . . 7 (Sect‘𝐶) = (Sect‘𝐶)
1713, 5, 1, 14, 15, 16invffval 17806 . . . . . 6 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))))
1817fneq1d 6662 . . . . 5 (𝜑 → (𝑁 Fn (𝐵 × 𝐵) ↔ (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) Fn (𝐵 × 𝐵)))
1912, 18mpbiri 258 . . . 4 (𝜑𝑁 Fn (𝐵 × 𝐵))
2014, 15opelxpd 5728 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
21 fvco2 7006 . . . 4 ((𝑁 Fn (𝐵 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵)) → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
2219, 20, 21syl2anc 584 . . 3 (𝜑 → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
23 df-ov 7434 . . 3 (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩)
24 ovex 7464 . . . . 5 (𝑋𝑁𝑌) ∈ V
25 dmeq 5917 . . . . . 6 (𝑧 = (𝑋𝑁𝑌) → dom 𝑧 = dom (𝑋𝑁𝑌))
26 eqid 2735 . . . . . 6 (𝑧 ∈ V ↦ dom 𝑧) = (𝑧 ∈ V ↦ dom 𝑧)
2724dmex 7932 . . . . . 6 dom (𝑋𝑁𝑌) ∈ V
2825, 26, 27fvmpt 7016 . . . . 5 ((𝑋𝑁𝑌) ∈ V → ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = dom (𝑋𝑁𝑌))
2924, 28ax-mp 5 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = dom (𝑋𝑁𝑌)
30 df-ov 7434 . . . . 5 (𝑋𝑁𝑌) = (𝑁‘⟨𝑋, 𝑌⟩)
3130fveq2i 6910 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
3229, 31eqtr3i 2765 . . 3 dom (𝑋𝑁𝑌) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
3322, 23, 323eqtr4g 2800 . 2 (𝜑 → (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = dom (𝑋𝑁𝑌))
348, 33eqtrd 2775 1 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  cop 4637  cmpt 5231   × cxp 5687  ccnv 5688  dom cdm 5689  ccom 5693   Fn wfn 6558  cfv 6563  (class class class)co 7431  cmpo 7433  Basecbs 17245  Catccat 17709  Sectcsect 17792  Invcinv 17793  Isociso 17794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-inv 17796  df-iso 17797
This theorem is referenced by:  inviso1  17814  invf  17816  invco  17819  dfiso2  17820  isohom  17824  oppciso  17829  cicsym  17852  ffthiso  17983  fuciso  18032  setciso  18145  catciso  18165  rngciso  20655  ringciso  20689  rngcisoALTV  48121  ringcisoALTV  48155
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