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

Theorem estrcval 18080
Description: Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
Hypotheses
Ref Expression
estrcval.c 𝐢 = (ExtStrCatβ€˜π‘ˆ)
estrcval.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
estrcval.h (πœ‘ β†’ 𝐻 = (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))
estrcval.o (πœ‘ β†’ Β· = (𝑣 ∈ (π‘ˆ Γ— π‘ˆ), 𝑧 ∈ π‘ˆ ↦ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓))))
Assertion
Ref Expression
estrcval (πœ‘ β†’ 𝐢 = {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
Distinct variable groups:   𝑓,𝑔,𝑣,π‘₯,𝑦,𝑧   πœ‘,𝑣,π‘₯,𝑦,𝑧   𝑣,π‘ˆ,π‘₯,𝑦,𝑧
Allowed substitution hints:   πœ‘(𝑓,𝑔)   𝐢(π‘₯,𝑦,𝑧,𝑣,𝑓,𝑔)   Β· (π‘₯,𝑦,𝑧,𝑣,𝑓,𝑔)   π‘ˆ(𝑓,𝑔)   𝐻(π‘₯,𝑦,𝑧,𝑣,𝑓,𝑔)   𝑉(π‘₯,𝑦,𝑧,𝑣,𝑓,𝑔)

Proof of Theorem estrcval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 estrcval.c . 2 𝐢 = (ExtStrCatβ€˜π‘ˆ)
2 df-estrc 18079 . . 3 ExtStrCat = (𝑒 ∈ V ↦ {⟨(Baseβ€˜ndx), π‘’βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓)))⟩})
3 simpr 484 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ 𝑒 = π‘ˆ)
43opeq2d 4880 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ⟨(Baseβ€˜ndx), π‘’βŸ© = ⟨(Baseβ€˜ndx), π‘ˆβŸ©)
5 eqidd 2732 . . . . . . 7 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) = ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))
63, 3, 5mpoeq123dv 7487 . . . . . 6 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) = (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))
7 estrcval.h . . . . . . 7 (πœ‘ β†’ 𝐻 = (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))
87adantr 480 . . . . . 6 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ 𝐻 = (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))
96, 8eqtr4d 2774 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) = 𝐻)
109opeq2d 4880 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))⟩ = ⟨(Hom β€˜ndx), 𝐻⟩)
113sqxpeqd 5708 . . . . . . 7 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (𝑒 Γ— 𝑒) = (π‘ˆ Γ— π‘ˆ))
12 eqidd 2732 . . . . . . 7 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓)))
1311, 3, 12mpoeq123dv 7487 . . . . . 6 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (π‘ˆ Γ— π‘ˆ), 𝑧 ∈ π‘ˆ ↦ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓))))
14 estrcval.o . . . . . . 7 (πœ‘ β†’ Β· = (𝑣 ∈ (π‘ˆ Γ— π‘ˆ), 𝑧 ∈ π‘ˆ ↦ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓))))
1514adantr 480 . . . . . 6 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ Β· = (𝑣 ∈ (π‘ˆ Γ— π‘ˆ), 𝑧 ∈ π‘ˆ ↦ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓))))
1613, 15eqtr4d 2774 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓))) = Β· )
1716opeq2d 4880 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ⟨(compβ€˜ndx), (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓)))⟩ = ⟨(compβ€˜ndx), Β· ⟩)
184, 10, 17tpeq123d 4752 . . 3 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ {⟨(Baseβ€˜ndx), π‘’βŸ©, ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))⟩, ⟨(compβ€˜ndx), (𝑣 ∈ (𝑒 Γ— 𝑒), 𝑧 ∈ 𝑒 ↦ (𝑔 ∈ ((Baseβ€˜π‘§) ↑m (Baseβ€˜(2nd β€˜π‘£))), 𝑓 ∈ ((Baseβ€˜(2nd β€˜π‘£)) ↑m (Baseβ€˜(1st β€˜π‘£))) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
19 estrcval.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
2019elexd 3494 . . 3 (πœ‘ β†’ π‘ˆ ∈ V)
21 tpex 7738 . . . 4 {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩} ∈ V
2221a1i 11 . . 3 (πœ‘ β†’ {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩} ∈ V)
232, 18, 20, 22fvmptd2 7006 . 2 (πœ‘ β†’ (ExtStrCatβ€˜π‘ˆ) = {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
241, 23eqtrid 2783 1 (πœ‘ β†’ 𝐢 = {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473  {ctp 4632  βŸ¨cop 4634   Γ— cxp 5674   ∘ ccom 5680  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414  1st c1st 7977  2nd c2nd 7978   ↑m cmap 8824  ndxcnx 17131  Basecbs 17149  Hom chom 17213  compcco 17214  ExtStrCatcestrc 18078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-oprab 7416  df-mpo 7417  df-estrc 18079
This theorem is referenced by:  estrcbas  18081  estrchomfval  18082  estrccofval  18085  dfrngc2  46959  dfringc2  47005
  Copyright terms: Public domain W3C validator