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Theorem monfval 17776
Description: Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
ismon.b 𝐵 = (Base‘𝐶)
ismon.h 𝐻 = (Hom ‘𝐶)
ismon.o · = (comp‘𝐶)
ismon.s 𝑀 = (Mono‘𝐶)
ismon.c (𝜑𝐶 ∈ Cat)
Assertion
Ref Expression
monfval (𝜑𝑀 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝑧,𝐵   𝜑,𝑓,𝑔,𝑥,𝑦,𝑧   𝐶,𝑓,𝑔,𝑥,𝑦,𝑧   𝑓,𝐻,𝑔,𝑥,𝑦,𝑧   · ,𝑓,𝑔,𝑥,𝑦,𝑧   𝑓,𝑀
Allowed substitution hints:   𝑀(𝑥,𝑦,𝑧,𝑔)

Proof of Theorem monfval
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismon.s . 2 𝑀 = (Mono‘𝐶)
2 ismon.c . . 3 (𝜑𝐶 ∈ Cat)
3 fvexd 6921 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6906 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 ismon.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5eqtr4di 2795 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 fvexd 6921 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8 simpl 482 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
98fveq2d 6910 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10 ismon.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
119, 10eqtr4di 2795 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12 simplr 769 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑏 = 𝐵)
13 simpr 484 . . . . . . . . 9 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → = 𝐻)
1413oveqd 7448 . . . . . . . 8 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑥𝑦) = (𝑥𝐻𝑦))
1513oveqd 7448 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑧𝑥) = (𝑧𝐻𝑥))
16 simpll 767 . . . . . . . . . . . . . . . 16 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑐 = 𝐶)
1716fveq2d 6910 . . . . . . . . . . . . . . 15 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18 ismon.o . . . . . . . . . . . . . . 15 · = (comp‘𝐶)
1917, 18eqtr4di 2795 . . . . . . . . . . . . . 14 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = · )
2019oveqd 7448 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦) = (⟨𝑧, 𝑥· 𝑦))
2120oveqd 7448 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔) = (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))
2215, 21mpteq12dv 5233 . . . . . . . . . . 11 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)))
2322cnveqd 5886 . . . . . . . . . 10 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)))
2423funeqd 6588 . . . . . . . . 9 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (Fun (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) ↔ Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))))
2512, 24raleqbidv 3346 . . . . . . . 8 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (∀𝑧𝑏 Fun (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) ↔ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))))
2614, 25rabeqbidv 3455 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → {𝑓 ∈ (𝑥𝑦) ∣ ∀𝑧𝑏 Fun (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))} = {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))})
2712, 12, 26mpoeq123dv 7508 . . . . . 6 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑥𝑏, 𝑦𝑏 ↦ {𝑓 ∈ (𝑥𝑦) ∣ ∀𝑧𝑏 Fun (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
287, 11, 27csbied2 3936 . . . . 5 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) / (𝑥𝑏, 𝑦𝑏 ↦ {𝑓 ∈ (𝑥𝑦) ∣ ∀𝑧𝑏 Fun (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
293, 6, 28csbied2 3936 . . . 4 (𝑐 = 𝐶(Base‘𝑐) / 𝑏(Hom ‘𝑐) / (𝑥𝑏, 𝑦𝑏 ↦ {𝑓 ∈ (𝑥𝑦) ∣ ∀𝑧𝑏 Fun (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
30 df-mon 17774 . . . 4 Mono = (𝑐 ∈ Cat ↦ (Base‘𝑐) / 𝑏(Hom ‘𝑐) / (𝑥𝑏, 𝑦𝑏 ↦ {𝑓 ∈ (𝑥𝑦) ∣ ∀𝑧𝑏 Fun (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}))
315fvexi 6920 . . . . 5 𝐵 ∈ V
3231, 31mpoex 8104 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}) ∈ V
3329, 30, 32fvmpt 7016 . . 3 (𝐶 ∈ Cat → (Mono‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
342, 33syl 17 . 2 (𝜑 → (Mono‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
351, 34eqtrid 2789 1 (𝜑𝑀 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  csb 3899  cop 4632  cmpt 5225  ccnv 5684  Fun wfun 6555  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Monocmon 17772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-mon 17774
This theorem is referenced by:  ismon  17777  monpropd  17781
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