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Theorem isoval 17809
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 17801 . . . . 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 5871 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶))
73, 4, 63eqtr4g 2802 . . 3 (𝜑𝐼 = ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁))
87oveqd 7448 . 2 (𝜑 → (𝑋𝐼𝑌) = (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌))
9 eqid 2737 . . . . . 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 5317 . . . . . 6 ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)) ∈ V
129, 11fnmpoi 8095 . . . . 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 2737 . . . . . . 7 (Sect‘𝐶) = (Sect‘𝐶)
1713, 5, 1, 14, 15, 16invffval 17802 . . . . . 6 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))))
1817fneq1d 6661 . . . . 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 5724 . . . 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 5914 . . . . . 6 (𝑧 = (𝑋𝑁𝑌) → dom 𝑧 = dom (𝑋𝑁𝑌))
26 eqid 2737 . . . . . 6 (𝑧 ∈ V ↦ dom 𝑧) = (𝑧 ∈ V ↦ dom 𝑧)
2724dmex 7931 . . . . . 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 6909 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
3229, 31eqtr3i 2767 . . 3 dom (𝑋𝑁𝑌) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
3322, 23, 323eqtr4g 2802 . 2 (𝜑 → (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = dom (𝑋𝑁𝑌))
348, 33eqtrd 2777 1 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  cop 4632  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  ccom 5689   Fn wfn 6556  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  Catccat 17707  Sectcsect 17788  Invcinv 17789  Isociso 17790
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-inv 17792  df-iso 17793
This theorem is referenced by:  inviso1  17810  invf  17812  invco  17815  dfiso2  17816  isohom  17820  oppciso  17825  cicsym  17848  ffthiso  17976  fuciso  18023  setciso  18136  catciso  18156  rngciso  20638  ringciso  20672  rngcisoALTV  48193  ringcisoALTV  48227
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