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Theorem monfval 17718
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 6911 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6896 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 ismon.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5eqtr4di 2783 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 fvexd 6911 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8 simpl 481 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
98fveq2d 6900 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10 ismon.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
119, 10eqtr4di 2783 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12 simplr 767 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑏 = 𝐵)
13 simpr 483 . . . . . . . . 9 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → = 𝐻)
1413oveqd 7436 . . . . . . . 8 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑥𝑦) = (𝑥𝐻𝑦))
1513oveqd 7436 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑧𝑥) = (𝑧𝐻𝑥))
16 simpll 765 . . . . . . . . . . . . . . . 16 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑐 = 𝐶)
1716fveq2d 6900 . . . . . . . . . . . . . . 15 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18 ismon.o . . . . . . . . . . . . . . 15 · = (comp‘𝐶)
1917, 18eqtr4di 2783 . . . . . . . . . . . . . 14 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = · )
2019oveqd 7436 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦) = (⟨𝑧, 𝑥· 𝑦))
2120oveqd 7436 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔) = (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))
2215, 21mpteq12dv 5240 . . . . . . . . . . 11 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)))
2322cnveqd 5878 . . . . . . . . . 10 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)))
2423funeqd 6576 . . . . . . . . 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 3329 . . . . . . . 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 3436 . . . . . . 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 7495 . . . . . 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 3929 . . . . 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 3929 . . . 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 17716 . . . 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 6910 . . . . 5 𝐵 ∈ V
3231, 31mpoex 8084 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}) ∈ V
3329, 30, 32fvmpt 7004 . . 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 2777 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 394   = wceq 1533  wcel 2098  wral 3050  {crab 3418  Vcvv 3461  csb 3889  cop 4636  cmpt 5232  ccnv 5677  Fun wfun 6543  cfv 6549  (class class class)co 7419  cmpo 7421  Basecbs 17183  Hom chom 17247  compcco 17248  Catccat 17647  Monocmon 17714
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-mon 17716
This theorem is referenced by:  ismon  17719  monpropd  17723
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