Theorem ismon 17512

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 17511	. . . 4 ⊢ (𝜑 → 𝑀 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑓 ∈ (𝑥𝐻𝑦) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔))}))
7		simprl 768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑥 = 𝑋)
8		simprr 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → 𝑦 = 𝑌)
9	7, 8	oveq12d 7331	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
10	7	oveq2d 7329	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑧𝐻𝑥) = (𝑧𝐻𝑋))
11	7	opeq2d 4820	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑋⟩)
12	11, 8	oveq12d 7331	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (⟨𝑧, 𝑥⟩ · 𝑦) = (⟨𝑧, 𝑋⟩ · 𝑌))
13	12	oveqd 7330	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔) = (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))
14	10, 13	mpteq12dv 5176	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → (𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))
15	14	cnveqd 5802	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ◡(𝑔 ∈ (𝑧𝐻𝑥) ↦ (𝑓(⟨𝑧, 𝑥⟩ · 𝑦)𝑔)) = ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))
16	15	funeqd 6490	. . . . . 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 3171	. . . . 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 3419	. . . 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 7346	. . . . . 6 ⊢ (𝑋𝐻𝑌) ∈ V
22	21	rabex 5269	. . . . 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	ovmpod 7463	. . 3 ⊢ (𝜑 → (𝑋𝑀𝑌) = {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))})
25	24	eleq2d 2823	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ 𝐹 ∈ {𝑓 ∈ (𝑋𝐻𝑌) ∣ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))}))
26		oveq1 7320	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))
27	26	mpteq2dv 5187	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)) = (𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))
28	27	cnveqd 5802	. . . . 5 ⊢ (𝑓 = 𝐹 → ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)) = ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))
29	28	funeqd 6490	. . . 4 ⊢ (𝑓 = 𝐹 → (Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝑓(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)) ↔ Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔))))
30	29	ralbidv 3171	. . 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 3633	. 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	bitrdi 286	1 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 Fun ◡(𝑔 ∈ (𝑧𝐻𝑋) ↦ (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3062  {crab 3404  Vcvv 3441  ⟨cop 4575   ↦ cmpt 5168  ^◡ccnv 5604  Fun wfun 6457  ‘cfv 6463  (class class class)co 7313  Basecbs 16979  Hom chom 17040  compcco 17041  Catccat 17440  Monocmon 17507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-1st 7874  df-2nd 7875  df-mon 17509

This theorem is referenced by:  ismon2  17513  monhom  17514  isepi  17519