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Theorem estrcval 18003
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 18002 . . 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 485 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ 𝑒 = π‘ˆ)
43opeq2d 4835 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ⟨(Baseβ€˜ndx), π‘’βŸ© = ⟨(Baseβ€˜ndx), π‘ˆβŸ©)
5 eqidd 2737 . . . . . . 7 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)) = ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))
63, 3, 5mpoeq123dv 7428 . . . . . 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 481 . . . . . 6 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ 𝐻 = (π‘₯ ∈ π‘ˆ, 𝑦 ∈ π‘ˆ ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))))
96, 8eqtr4d 2779 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯))) = 𝐻)
109opeq2d 4835 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ⟨(Hom β€˜ndx), (π‘₯ ∈ 𝑒, 𝑦 ∈ 𝑒 ↦ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))⟩ = ⟨(Hom β€˜ndx), 𝐻⟩)
113sqxpeqd 5663 . . . . . . 7 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (𝑒 Γ— 𝑒) = (π‘ˆ Γ— π‘ˆ))
12 eqidd 2737 . . . . . . 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 7428 . . . . . 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 481 . . . . . 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 2779 . . . . 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 4835 . . . 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 4707 . . 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 3463 . . 3 (πœ‘ β†’ π‘ˆ ∈ V)
21 tpex 7677 . . . 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 6953 . 2 (πœ‘ β†’ (ExtStrCatβ€˜π‘ˆ) = {⟨(Baseβ€˜ndx), π‘ˆβŸ©, ⟨(Hom β€˜ndx), 𝐻⟩, ⟨(compβ€˜ndx), Β· ⟩})
241, 23eqtrid 2788 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 396   = wceq 1541   ∈ wcel 2106  Vcvv 3443  {ctp 4588  βŸ¨cop 4590   Γ— cxp 5629   ∘ ccom 5635  β€˜cfv 6493  (class class class)co 7353   ∈ cmpo 7355  1st c1st 7915  2nd c2nd 7916   ↑m cmap 8761  ndxcnx 17057  Basecbs 17075  Hom chom 17136  compcco 17137  ExtStrCatcestrc 18001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-oprab 7357  df-mpo 7358  df-estrc 18002
This theorem is referenced by:  estrcbas  18004  estrchomfval  18005  estrccofval  18008  dfrngc2  46202  dfringc2  46248
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