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	⊢ (𝜑 → 𝐶 ∈ Cat)

Assertion

Ref	Expression
monfval	⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 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 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismon.s	. 2 ⊢ 𝑀 = (Mono‘𝐶)
2		ismon.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 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)
6	4, 5	eqtr4di 2783	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7		fvexd 6911	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8		simpl 481	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
9	8	fveq2d 6900	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10		ismon.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
11	9, 10	eqtr4di 2783	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12		simplr 767	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑏 = 𝐵)
13		simpr 483	. . . . . . . . 9 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ℎ = 𝐻)
14	13	oveqd 7436	. . . . . . . 8 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
15	13	oveqd 7436	. . . . . . . . . . . 12 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑧ℎ𝑥) = (𝑧𝐻𝑥))
16		simpll 765	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑐 = 𝐶)
17	16	fveq2d 6900	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18		ismon.o	. . . . . . . . . . . . . . 15 ⊢ · = (comp‘𝐶)
19	17, 18	eqtr4di 2783	. . . . . . . . . . . . . 14 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = · )
20	19	oveqd 7436	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦) = (⟨𝑧, 𝑥⟩ · 𝑦))
21	20	oveqd 7436	. . . . . . . . . . . 12 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔) = (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))
22	15, 21	mpteq12dv 5240	. . . . . . . . . . 11 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔)))
23	22	cnveqd 5878	. . . . . . . . . 10 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) = ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔)))
24	23	funeqd 6576	. . . . . . . . 9 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) ↔ Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))))
25	12, 24	raleqbidv 3329	. . . . . . . 8 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔)) ↔ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))))
26	14, 25	rabeqbidv 3436	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))} = {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))})
27	12, 12, 26	mpoeq123dv 7495	. . . . . 6 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}))
28	7, 11, 27	csbied2 3929	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → ⦋(Hom ‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}))
29	3, 6, 28	csbied2 3929	. . . 4 ⊢ (𝑐 = 𝐶 → ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}))
30		df-mon 17716	. . . 4 ⊢ Mono = (𝑐 ∈ Cat ↦ ⦋(Base‘𝑐) / 𝑏⦌⦋(Hom ‘𝑐) / ℎ⦌(𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ {𝑓 ∈ (𝑥ℎ𝑦) ∣ ∀𝑧 ∈ 𝑏 Fun ◡(𝑔 ∈ (𝑧ℎ𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔))}))
31	5	fvexi 6910	. . . . 5 ⊢ 𝐵 ∈ V
32	31, 31	mpoex 8084	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}) ∈ V
33	29, 30, 32	fvmpt 7004	. . 3 ⊢ (𝐶 ∈ Cat → (Mono‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}))
34	2, 33	syl 17	. 2 ⊢ (𝜑 → (Mono‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}))
35	1, 34	eqtrid 2777	1 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}))

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