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

Theorem termoval 17944
Description: The value of the terminal object function, i.e. the set of all terminal objects of a category. (Contributed by AV, 3-Apr-2020.)
Hypotheses
Ref Expression
initoval.c (πœ‘ β†’ 𝐢 ∈ Cat)
initoval.b 𝐡 = (Baseβ€˜πΆ)
initoval.h 𝐻 = (Hom β€˜πΆ)
Assertion
Ref Expression
termoval (πœ‘ β†’ (TermOβ€˜πΆ) = {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘π»π‘Ž)})
Distinct variable groups:   π‘Ž,𝑏,β„Ž   𝐡,π‘Ž,𝑏   𝐢,π‘Ž,𝑏,β„Ž
Allowed substitution hints:   πœ‘(β„Ž,π‘Ž,𝑏)   𝐡(β„Ž)   𝐻(β„Ž,π‘Ž,𝑏)

Proof of Theorem termoval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-termo 17935 . 2 TermO = (𝑐 ∈ Cat ↦ {π‘Ž ∈ (Baseβ€˜π‘) ∣ βˆ€π‘ ∈ (Baseβ€˜π‘)βˆƒ!β„Ž β„Ž ∈ (𝑏(Hom β€˜π‘)π‘Ž)})
2 fveq2 6892 . . . 4 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
3 initoval.b . . . 4 𝐡 = (Baseβ€˜πΆ)
42, 3eqtr4di 2791 . . 3 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = 𝐡)
5 fveq2 6892 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = (Hom β€˜πΆ))
6 initoval.h . . . . . . . 8 𝐻 = (Hom β€˜πΆ)
75, 6eqtr4di 2791 . . . . . . 7 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = 𝐻)
87oveqd 7426 . . . . . 6 (𝑐 = 𝐢 β†’ (𝑏(Hom β€˜π‘)π‘Ž) = (π‘π»π‘Ž))
98eleq2d 2820 . . . . 5 (𝑐 = 𝐢 β†’ (β„Ž ∈ (𝑏(Hom β€˜π‘)π‘Ž) ↔ β„Ž ∈ (π‘π»π‘Ž)))
109eubidv 2581 . . . 4 (𝑐 = 𝐢 β†’ (βˆƒ!β„Ž β„Ž ∈ (𝑏(Hom β€˜π‘)π‘Ž) ↔ βˆƒ!β„Ž β„Ž ∈ (π‘π»π‘Ž)))
114, 10raleqbidv 3343 . . 3 (𝑐 = 𝐢 β†’ (βˆ€π‘ ∈ (Baseβ€˜π‘)βˆƒ!β„Ž β„Ž ∈ (𝑏(Hom β€˜π‘)π‘Ž) ↔ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘π»π‘Ž)))
124, 11rabeqbidv 3450 . 2 (𝑐 = 𝐢 β†’ {π‘Ž ∈ (Baseβ€˜π‘) ∣ βˆ€π‘ ∈ (Baseβ€˜π‘)βˆƒ!β„Ž β„Ž ∈ (𝑏(Hom β€˜π‘)π‘Ž)} = {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘π»π‘Ž)})
13 initoval.c . 2 (πœ‘ β†’ 𝐢 ∈ Cat)
143fvexi 6906 . . . 4 𝐡 ∈ V
1514rabex 5333 . . 3 {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘π»π‘Ž)} ∈ V
1615a1i 11 . 2 (πœ‘ β†’ {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘π»π‘Ž)} ∈ V)
171, 12, 13, 16fvmptd3 7022 1 (πœ‘ β†’ (TermOβ€˜πΆ) = {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘π»π‘Ž)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆƒ!weu 2563  βˆ€wral 3062  {crab 3433  Vcvv 3475  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Hom chom 17208  Catccat 17608  TermOctermo 17932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-termo 17935
This theorem is referenced by:  istermo  17947  istermoi  17950  dfinito2  17953
  Copyright terms: Public domain W3C validator