Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  isoval Structured version   Visualization version   GIF version

Theorem isoval 16248
 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 16240 . . . . 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 5204 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁) = ((𝑧 ∈ V ↦ dom 𝑧) ∘ (Inv‘𝐶))
73, 4, 63eqtr4g 2669 . . 3 (𝜑𝐼 = ((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁))
87oveqd 6566 . 2 (𝜑 → (𝑋𝐼𝑌) = (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌))
9 eqid 2610 . . . . . 6 (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)))
10 ovex 6577 . . . . . . 7 (𝑥(Sect‘𝐶)𝑦) ∈ V
1110inex1 4727 . . . . . 6 ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥)) ∈ V
129, 11fnmpt2i 7128 . . . . 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 2610 . . . . . . 7 (Sect‘𝐶) = (Sect‘𝐶)
1713, 5, 1, 14, 15, 16invffval 16241 . . . . . 6 (𝜑𝑁 = (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))))
1817fneq1d 5895 . . . . 5 (𝜑 → (𝑁 Fn (𝐵 × 𝐵) ↔ (𝑥𝐵, 𝑦𝐵 ↦ ((𝑥(Sect‘𝐶)𝑦) ∩ (𝑦(Sect‘𝐶)𝑥))) Fn (𝐵 × 𝐵)))
1912, 18mpbiri 247 . . . 4 (𝜑𝑁 Fn (𝐵 × 𝐵))
20 opelxpi 5072 . . . . 5 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
2114, 15, 20syl2anc 691 . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
22 fvco2 6183 . . . 4 ((𝑁 Fn (𝐵 × 𝐵) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵)) → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
2319, 21, 22syl2anc 691 . . 3 (𝜑 → (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩)))
24 df-ov 6552 . . 3 (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = (((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)‘⟨𝑋, 𝑌⟩)
25 ovex 6577 . . . . 5 (𝑋𝑁𝑌) ∈ V
26 dmeq 5246 . . . . . 6 (𝑧 = (𝑋𝑁𝑌) → dom 𝑧 = dom (𝑋𝑁𝑌))
27 eqid 2610 . . . . . 6 (𝑧 ∈ V ↦ dom 𝑧) = (𝑧 ∈ V ↦ dom 𝑧)
2825dmex 6991 . . . . . 6 dom (𝑋𝑁𝑌) ∈ V
2926, 27, 28fvmpt 6191 . . . . 5 ((𝑋𝑁𝑌) ∈ V → ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = dom (𝑋𝑁𝑌))
3025, 29ax-mp 5 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = dom (𝑋𝑁𝑌)
31 df-ov 6552 . . . . 5 (𝑋𝑁𝑌) = (𝑁‘⟨𝑋, 𝑌⟩)
3231fveq2i 6106 . . . 4 ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑋𝑁𝑌)) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
3330, 32eqtr3i 2634 . . 3 dom (𝑋𝑁𝑌) = ((𝑧 ∈ V ↦ dom 𝑧)‘(𝑁‘⟨𝑋, 𝑌⟩))
3423, 24, 333eqtr4g 2669 . 2 (𝜑 → (𝑋((𝑧 ∈ V ↦ dom 𝑧) ∘ 𝑁)𝑌) = dom (𝑋𝑁𝑌))
358, 34eqtrd 2644 1 (𝜑 → (𝑋𝐼𝑌) = dom (𝑋𝑁𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  dom cdm 5038   ∘ ccom 5042   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  Catccat 16148  Sectcsect 16227  Invcinv 16228  Isociso 16229 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-inv 16231  df-iso 16232 This theorem is referenced by:  inviso1  16249  invf  16251  invco  16254  dfiso2  16255  isohom  16259  oppciso  16264  cicsym  16287  funciso  16357  ffthiso  16412  fuciso  16458  setciso  16564  catciso  16580  rngciso  41774  rngcisoALTV  41786  ringciso  41825  ringcisoALTV  41849
 Copyright terms: Public domain W3C validator