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Theorem initoval 17945
Description: The value of the initial object function, i.e. the set of all initial 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
initoval (πœ‘ β†’ (InitOβ€˜πΆ) = {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘Žπ»π‘)})
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 initoval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 df-inito 17936 . 2 InitO = (𝑐 ∈ Cat ↦ {π‘Ž ∈ (Baseβ€˜π‘) ∣ βˆ€π‘ ∈ (Baseβ€˜π‘)βˆƒ!β„Ž β„Ž ∈ (π‘Ž(Hom β€˜π‘)𝑏)})
2 fveq2 6891 . . . 4 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
3 initoval.b . . . 4 𝐡 = (Baseβ€˜πΆ)
42, 3eqtr4di 2790 . . 3 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = 𝐡)
5 fveq2 6891 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = (Hom β€˜πΆ))
6 initoval.h . . . . . . . 8 𝐻 = (Hom β€˜πΆ)
75, 6eqtr4di 2790 . . . . . . 7 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = 𝐻)
87oveqd 7428 . . . . . 6 (𝑐 = 𝐢 β†’ (π‘Ž(Hom β€˜π‘)𝑏) = (π‘Žπ»π‘))
98eleq2d 2819 . . . . 5 (𝑐 = 𝐢 β†’ (β„Ž ∈ (π‘Ž(Hom β€˜π‘)𝑏) ↔ β„Ž ∈ (π‘Žπ»π‘)))
109eubidv 2580 . . . 4 (𝑐 = 𝐢 β†’ (βˆƒ!β„Ž β„Ž ∈ (π‘Ž(Hom β€˜π‘)𝑏) ↔ βˆƒ!β„Ž β„Ž ∈ (π‘Žπ»π‘)))
114, 10raleqbidv 3342 . . 3 (𝑐 = 𝐢 β†’ (βˆ€π‘ ∈ (Baseβ€˜π‘)βˆƒ!β„Ž β„Ž ∈ (π‘Ž(Hom β€˜π‘)𝑏) ↔ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘Žπ»π‘)))
124, 11rabeqbidv 3449 . 2 (𝑐 = 𝐢 β†’ {π‘Ž ∈ (Baseβ€˜π‘) ∣ βˆ€π‘ ∈ (Baseβ€˜π‘)βˆƒ!β„Ž β„Ž ∈ (π‘Ž(Hom β€˜π‘)𝑏)} = {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘Žπ»π‘)})
13 initoval.c . 2 (πœ‘ β†’ 𝐢 ∈ Cat)
143fvexi 6905 . . . 4 𝐡 ∈ V
1514rabex 5332 . . 3 {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘Žπ»π‘)} ∈ V
1615a1i 11 . 2 (πœ‘ β†’ {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘Žπ»π‘)} ∈ V)
171, 12, 13, 16fvmptd3 7021 1 (πœ‘ β†’ (InitOβ€˜πΆ) = {π‘Ž ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (π‘Žπ»π‘)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  βˆƒ!weu 2562  βˆ€wral 3061  {crab 3432  Vcvv 3474  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  Hom chom 17210  Catccat 17610  InitOcinito 17933
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  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-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-ov 7414  df-inito 17936
This theorem is referenced by:  isinito  17948  isinitoi  17951  dftermo2  17956
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