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

Theorem estrcval 18079
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 18078 . . 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 483 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ 𝑒 = π‘ˆ)
43opeq2d 4879 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ⟨(Baseβ€˜ndx), π‘’βŸ© = ⟨(Baseβ€˜ndx), π‘ˆβŸ©)
5 eqidd 2731 . . . . . . 7 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) = ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))
63, 3, 5mpoeq123dv 7486 . . . . . 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 479 . . . . . 6 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ 𝐻 = (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))
96, 8eqtr4d 2773 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) = 𝐻)
109opeq2d 4879 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))⟩ = ⟨(Hom β€˜ndx), 𝐻⟩)
113sqxpeqd 5707 . . . . . . 7 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (𝑒 Γ— 𝑒) = (π‘ˆ Γ— π‘ˆ))
12 eqidd 2731 . . . . . . 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 7486 . . . . . 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 479 . . . . . 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 2773 . . . . 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 4879 . . . 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 4751 . . 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 3493 . . 3 (πœ‘ β†’ π‘ˆ ∈ V)
21 tpex 7736 . . . 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 7005 . 2 (πœ‘ β†’ (ExtStrCatβ€˜π‘ˆ) = {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
241, 23eqtrid 2782 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 394   = wceq 1539   ∈ wcel 2104  Vcvv 3472  {ctp 4631  βŸ¨cop 4633   Γ— cxp 5673   ∘ ccom 5679  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976   ↑m cmap 8822  ndxcnx 17130  Basecbs 17148  Hom chom 17212  compcco 17213  ExtStrCatcestrc 18077
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-oprab 7415  df-mpo 7416  df-estrc 18078
This theorem is referenced by:  estrcbas  18080  estrchomfval  18081  estrccofval  18084  dfrngc2  46958  dfringc2  47004
  Copyright terms: Public domain W3C validator