Theorem ismon 16393

Description: Definition of a monomorphism in a category. (Contributed by Mario Carneiro, 2-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)
ismon.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ismon.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
ismon	⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))))

Distinct variable groups:   𝑧,𝑔,𝐵   𝜑,𝑔,𝑧   𝐶,𝑔,𝑧   𝑔,𝐻,𝑧   · ,𝑔,𝑧   𝑔,𝐹,𝑧   𝑔,𝑋,𝑧   𝑔,𝑌,𝑧

Allowed substitution hints:   𝑀(𝑧,𝑔)

Proof of Theorem ismon

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismon.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
2		ismon.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
3		ismon.o	. . . . 5 ⊢ · = (comp‘𝐶)
4		ismon.s	. . . . 5 ⊢ 𝑀 = (Mono‘𝐶)
5		ismon.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat)
6	1, 2, 3, 4, 5	monfval 16392	. . . 4 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}))
7		simprl 794	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
8		simprr 796	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
9	7, 8	oveq12d 6668	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
10	7	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑧𝐻𝑥) = (𝑧𝐻𝑋))
11	7	opeq2d 4409	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑋⟩)
12	11, 8	oveq12d 6668	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (⟨𝑧, 𝑥⟩ · 𝑦) = (⟨𝑧, 𝑋⟩ · 𝑌))
13	12	oveqd 6667	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔) = (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))
14	10, 13	mpteq12dv 4733	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))
15	14	cnveqd 5298	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔)) = ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))
16	15	funeqd 5910	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔)) ↔ Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))))
17	16	ralbidv 2986	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔)) ↔ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))))
18	9, 17	rabeqbidv 3195	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))} = {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))})
19		ismon.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
20		ismon.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
21		ovex 6678	. . . . . 6 ⊢ (𝑋𝐻𝑌) ∈ V
22	21	rabex 4813	. . . . 5 ⊢ {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))} ∈ V
23	22	a1i 11	. . . 4 ⊢ (𝜑 → {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))} ∈ V)
24	6, 18, 19, 20, 23	ovmpt2d 6788	. . 3 ⊢ (𝜑 → (𝑋𝑀𝑌) = {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))})
25	24	eleq2d 2687	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹 ∈ {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))}))
26		oveq1 6657	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))
27	26	mpteq2dv 4745	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))
28	27	cnveqd 5298	. . . . 5 ⊢ (𝑓 = 𝐹 → ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)) = ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))
29	28	funeqd 5910	. . . 4 ⊢ (𝑓 = 𝐹 → (Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)) ↔ Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))))
30	29	ralbidv 2986	. . 3 ⊢ (𝑓 = 𝐹 → (∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)) ↔ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))))
31	30	elrab 3363	. 2 ⊢ (𝐹 ∈ {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))} ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))))
32	25, 31	syl6bb 276	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200  ⟨cop 4183   ↦ cmpt 4729  ^◡ccnv 5113  Fun wfun 5882  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Hom chom 15952  compcco 15953  Catccat 16325  Monocmon 16388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-mon 16390

This theorem is referenced by:  ismon2  16394  monhom  16395  isepi  16400