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Theorem istermoi 17043
 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 17037 . . . . 5 (𝜑 → (TermO‘𝐶) = {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)})
54eleq2d 2845 . . . 4 (𝜑 → (𝑂 ∈ (TermO‘𝐶) ↔ 𝑂 ∈ {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)}))
6 elrabi 3567 . . . 4 (𝑂 ∈ {𝑎𝐵 ∣ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑎)} → 𝑂𝐵)
75, 6syl6bi 245 . . 3 (𝜑 → (𝑂 ∈ (TermO‘𝐶) → 𝑂𝐵))
87imp 397 . 2 ((𝜑𝑂 ∈ (TermO‘𝐶)) → 𝑂𝐵)
91adantr 474 . . . . 5 ((𝜑𝑂𝐵) → 𝐶 ∈ Cat)
10 simpr 479 . . . . 5 ((𝜑𝑂𝐵) → 𝑂𝐵)
112, 3, 9, 10istermo 17040 . . . 4 ((𝜑𝑂𝐵) → (𝑂 ∈ (TermO‘𝐶) ↔ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))
1211biimpd 221 . . 3 ((𝜑𝑂𝐵) → (𝑂 ∈ (TermO‘𝐶) → ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))
1312impancom 445 . 2 ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐵 → ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))
148, 13jcai 512 1 ((𝜑𝑂 ∈ (TermO‘𝐶)) → (𝑂𝐵 ∧ ∀𝑏𝐵 ∃! ∈ (𝑏𝐻𝑂)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∃!weu 2586  ∀wral 3090  {crab 3094  ‘cfv 6137  (class class class)co 6924  Basecbs 16259  Hom chom 16353  Catccat 16714  TermOctermo 17028 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927  df-termo 17031 This theorem is referenced by:  termoid  17045  termoo  17047  termoeu1  17057
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