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	⊢ (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 6857	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) ∈ V)
4		fveq2 6842	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
5		ismon.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
6	4, 5	eqtr4di 2794	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
7		fvexd 6857	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) ∈ V)
8		simpl 483	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
9	8	fveq2d 6846	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
10		ismon.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
11	9, 10	eqtr4di 2794	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
12		simplr 767	. . . . . . 7 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑏 = 𝐵)
13		simpr 485	. . . . . . . . 9 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → ℎ = 𝐻)
14	13	oveqd 7374	. . . . . . . 8 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑥ℎ𝑦) = (𝑥𝐻𝑦))
15	13	oveqd 7374	. . . . . . . . . . . 12 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑧ℎ𝑥) = (𝑧𝐻𝑥))
16		simpll 765	. . . . . . . . . . . . . . . 16 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → 𝑐 = 𝐶)
17	16	fveq2d 6846	. . . . . . . . . . . . . . 15 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18		ismon.o	. . . . . . . . . . . . . . 15 ⊢ · = (comp‘𝐶)
19	17, 18	eqtr4di 2794	. . . . . . . . . . . . . 14 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (comp‘𝑐) = · )
20	19	oveqd 7374	. . . . . . . . . . . . 13 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦) = (⟨𝑧, 𝑥⟩ · 𝑦))
21	20	oveqd 7374	. . . . . . . . . . . 12 ⊢ (((𝑐 = 𝐶 ∧ 𝑏 = 𝐵) ∧ ℎ = 𝐻) → (𝑓(⟨𝑧, 𝑥⟩(comp‘𝑐)𝑦)𝑔) = (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))
22	15, 21	mpteq12dv 5196	. . . . . . . . . . 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 5831	. . . . . . . . . 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 6523	. . . . . . . . 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 3319	. . . . . . . 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 3424	. . . . . . 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 7432	. . . . . 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 3895	. . . . 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 3895	. . . 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 17613	. . . 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 6856	. . . . 5 ⊢ 𝐵 ∈ V
32	31, 31	mpoex 8012	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}) ∈ V
33	29, 30, 32	fvmpt 6948	. . 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 2788	1 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}))

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