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Theorem isinito 17945
Description: The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.)
Hypotheses
Ref Expression
isinito.b 𝐡 = (Baseβ€˜πΆ)
isinito.h 𝐻 = (Hom β€˜πΆ)
isinito.c (πœ‘ β†’ 𝐢 ∈ Cat)
isinito.i (πœ‘ β†’ 𝐼 ∈ 𝐡)
Assertion
Ref Expression
isinito (πœ‘ β†’ (𝐼 ∈ (InitOβ€˜πΆ) ↔ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝐼𝐻𝑏)))
Distinct variable groups:   𝐡,𝑏   𝐢,𝑏,β„Ž   𝐼,𝑏,β„Ž
Allowed substitution hints:   πœ‘(β„Ž,𝑏)   𝐡(β„Ž)   𝐻(β„Ž,𝑏)

Proof of Theorem isinito
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 isinito.c . . . 4 (πœ‘ β†’ 𝐢 ∈ Cat)
2 isinito.b . . . 4 𝐡 = (Baseβ€˜πΆ)
3 isinito.h . . . 4 𝐻 = (Hom β€˜πΆ)
41, 2, 3initoval 17942 . . 3 (πœ‘ β†’ (InitOβ€˜πΆ) = {𝑖 ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝑖𝐻𝑏)})
54eleq2d 2811 . 2 (πœ‘ β†’ (𝐼 ∈ (InitOβ€˜πΆ) ↔ 𝐼 ∈ {𝑖 ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝑖𝐻𝑏)}))
6 isinito.i . . 3 (πœ‘ β†’ 𝐼 ∈ 𝐡)
7 oveq1 7408 . . . . . . 7 (𝑖 = 𝐼 β†’ (𝑖𝐻𝑏) = (𝐼𝐻𝑏))
87eleq2d 2811 . . . . . 6 (𝑖 = 𝐼 β†’ (β„Ž ∈ (𝑖𝐻𝑏) ↔ β„Ž ∈ (𝐼𝐻𝑏)))
98eubidv 2572 . . . . 5 (𝑖 = 𝐼 β†’ (βˆƒ!β„Ž β„Ž ∈ (𝑖𝐻𝑏) ↔ βˆƒ!β„Ž β„Ž ∈ (𝐼𝐻𝑏)))
109ralbidv 3169 . . . 4 (𝑖 = 𝐼 β†’ (βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝑖𝐻𝑏) ↔ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝐼𝐻𝑏)))
1110elrab3 3676 . . 3 (𝐼 ∈ 𝐡 β†’ (𝐼 ∈ {𝑖 ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝑖𝐻𝑏)} ↔ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝐼𝐻𝑏)))
126, 11syl 17 . 2 (πœ‘ β†’ (𝐼 ∈ {𝑖 ∈ 𝐡 ∣ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝑖𝐻𝑏)} ↔ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝐼𝐻𝑏)))
135, 12bitrd 279 1 (πœ‘ β†’ (𝐼 ∈ (InitOβ€˜πΆ) ↔ βˆ€π‘ ∈ 𝐡 βˆƒ!β„Ž β„Ž ∈ (𝐼𝐻𝑏)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  βˆƒ!weu 2554  βˆ€wral 3053  {crab 3424  β€˜cfv 6533  (class class class)co 7401  Basecbs 17140  Hom chom 17204  Catccat 17604  InitOcinito 17930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-inito 17933
This theorem is referenced by:  isinitoi  17948  initoeu2  17965  zrinitorngc  20523  zrninitoringc  20557  irinitoringc  21329
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