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Theorem istermoi 17239
Description: Implication of a class being a terminal object. (Contributed by AV, 18-Apr-2020.)
Hypotheses
Ref Expression
isinitoi.b 𝐵 = (Base‘𝐶)
isinitoi.h 𝐻 = (Hom ‘𝐶)
isinitoi.c (𝜑𝐶 ∈ Cat)
Assertion
Ref Expression
istermoi ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))
Distinct variable groups:   𝐵,𝑏   𝐶,𝑏,   𝑂,𝑏,
Allowed substitution hints:   𝜑(,𝑏)   𝐵()   𝐻(,𝑏)

Proof of Theorem istermoi
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 isinitoi.c . . . . . 6 (𝜑𝐶 ∈ Cat)
2 isinitoi.b . . . . . 6 𝐵 = (Base‘𝐶)
3 isinitoi.h . . . . . 6 𝐻 = (Hom ‘𝐶)
41, 2, 3termoval 17233 . . . . 5 (𝜑 → (TermO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
54eleq2d 2896 . . . 4 (𝜑 → (𝑂 ∈ (TermO‘𝐶) ↔ 𝑂 ∈ {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)}))
6 elrabi 3651 . . . 4 (𝑂 ∈ {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)} → 𝑂𝐵)
75, 6syl6bi 255 . . 3 (𝜑 → (𝑂 ∈ (TermO‘𝐶) → 𝑂𝐵))
87imp 409 . 2 ((𝜑𝑂 ∈ (TermO‘𝐶)) → 𝑂𝐵)
91adantr 483 . . . . 5 ((𝜑𝑂𝐵) → 𝐶 ∈ Cat)
10 simpr 487 . . . . 5 ((𝜑𝑂𝐵) → 𝑂𝐵)
112, 3, 9, 10istermo 17236 . . . 4 ((𝜑𝑂𝐵) → (𝑂 ∈ (TermO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))
1211biimpd 231 . . 3 ((𝜑𝑂𝐵) → (𝑂 ∈ (TermO‘𝐶) → ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))
1312impancom 454 . 2 ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐵 → ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))
148, 13jcai 519 1 ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  ∃!weu 2652  wral 3125  {crab 3129  cfv 6327  (class class class)co 7129  Basecbs 16458  Hom chom 16551  Catccat 16910  TermOctermo 17224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6286  df-fun 6329  df-fv 6335  df-ov 7132  df-termo 17227
This theorem is referenced by:  termoid  17241  termoo  17243  termoeu1  17253
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