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Theorem monfval 17615
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 6857 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4 fveq2 6842 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5 ismon.b . . . . . 6 𝐵 = (Base‘𝐶)
64, 5eqtr4di 2794 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7 fvexd 6857 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8 simpl 483 . . . . . . . 8 ((𝑐 = 𝐶𝑏 = 𝐵) → 𝑐 = 𝐶)
98fveq2d 6846 . . . . . . 7 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10 ismon.h . . . . . . 7 𝐻 = (Hom ‘𝐶)
119, 10eqtr4di 2794 . . . . . 6 ((𝑐 = 𝐶𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12 simplr 767 . . . . . . 7 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑏 = 𝐵)
13 simpr 485 . . . . . . . . 9 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → = 𝐻)
1413oveqd 7374 . . . . . . . 8 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑥𝑦) = (𝑥𝐻𝑦))
1513oveqd 7374 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑧𝑥) = (𝑧𝐻𝑥))
16 simpll 765 . . . . . . . . . . . . . . . 16 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → 𝑐 = 𝐶)
1716fveq2d 6846 . . . . . . . . . . . . . . 15 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18 ismon.o . . . . . . . . . . . . . . 15 · = (comp‘𝐶)
1917, 18eqtr4di 2794 . . . . . . . . . . . . . 14 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (comp‘𝑐) = · )
2019oveqd 7374 . . . . . . . . . . . . 13 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦) = (⟨𝑧, 𝑥· 𝑦))
2120oveqd 7374 . . . . . . . . . . . 12 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔) = (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))
2215, 21mpteq12dv 5196 . . . . . . . . . . 11 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)))
2322cnveqd 5831 . . . . . . . . . 10 (((𝑐 = 𝐶𝑏 = 𝐵) ∧ = 𝐻) → (𝑔 ∈ (𝑧𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔)))
2423funeqd 6523 . . . . . . . . 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 3319 . . . . . . . 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 3424 . . . . . . 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 7432 . . . . . 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 3895 . . . . 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 3895 . . . 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 17613 . . . 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 6856 . . . . 5 𝐵 ∈ V
3231, 31mpoex 8012 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧𝐵 Fun (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥· 𝑦)𝑔))}) ∈ V
3329, 30, 32fvmpt 6948 . . 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 2788 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 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  csb 3855  cop 4592  cmpt 5188  ccnv 5632  Fun wfun 6490  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  Hom chom 17144  compcco 17145  Catccat 17544  Monocmon 17611
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-mon 17613
This theorem is referenced by:  ismon  17616  monpropd  17620
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