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Theorem monfval 17002
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 6676 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6661 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 ismon.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5syl6eqr 2877 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 fvexd 6676 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8 simpl 486 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
98fveq2d 6665 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10 ismon.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
119, 10syl6eqr 2877 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12 simplr 768 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑏 = 𝐵)
13 simpr 488 . . . . . . . . 9 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → = 𝐻)
1413oveqd 7166 . . . . . . . 8 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑥𝑦) = (𝑥𝐻𝑦))
1513oveqd 7166 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑧𝑥) = (𝑧𝐻𝑥))
16 simpll 766 . . . . . . . . . . . . . . . 16 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑐 = 𝐶)
1716fveq2d 6665 . . . . . . . . . . . . . . 15 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18 ismon.o . . . . . . . . . . . . . . 15 · = (comp‘𝐶)
1917, 18syl6eqr 2877 . . . . . . . . . . . . . 14 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = · )
2019oveqd 7166 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦) = (⟨𝑧, 𝑥· 𝑦))
2120oveqd 7166 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔) = (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))
2215, 21mpteq12dv 5137 . . . . . . . . . . 11 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)))
2322cnveqd 5733 . . . . . . . . . 10 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)))
2423funeqd 6365 . . . . . . . . 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 3392 . . . . . . . 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 3471 . . . . . . 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 7222 . . . . . 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 3903 . . . . 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 3903 . . . 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 17000 . . . 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 6675 . . . . 5 𝐵 ∈ V
3231, 31mpoex 7773 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}) ∈ V
3329, 30, 32fvmpt 6759 . . 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, 34syl5eq 2871 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 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137  Vcvv 3480  csb 3866  cop 4556  cmpt 5132  ccnv 5541  Fun wfun 6337  cfv 6343  (class class class)co 7149  cmpo 7151  Basecbs 16483  Hom chom 16576  compcco 16577  Catccat 16935  Monocmon 16998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-mon 17000
This theorem is referenced by:  ismon  17003  monpropd  17007
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