Theorem moni 16396

Description: Property of a monomorphism. (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)
moni.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
moni.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝑀𝑌))
moni.g	⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋))
moni.k	⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋))

Assertion

Ref	Expression
moni	⊢ (𝜑 → ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) ↔ 𝐺 = 𝐾))

Proof of Theorem moni

Dummy variables 𝑔 ℎ 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		moni.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝑀𝑌))
2		ismon.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
3		ismon.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
4		ismon.o	. . . . . 6 ⊢ · = (comp‘𝐶)
5		ismon.s	. . . . . 6 ⊢ 𝑀 = (Mono‘𝐶)
6		ismon.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		ismon.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		ismon.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	2, 3, 4, 5, 6, 7, 8	ismon2 16394	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝑀𝑌) ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) → 𝑔 = ℎ))))
10	1, 9	mpbid 222	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) → 𝑔 = ℎ)))
11	10	simprd 479	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) → 𝑔 = ℎ))
12		moni.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
13		moni.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (𝑍𝐻𝑋))
14	13	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑍𝐻𝑋))
15		simpr 477	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍)
16	15	oveq1d 6665	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝑧𝐻𝑋) = (𝑍𝐻𝑋))
17	14, 16	eleqtrrd 2704	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑧𝐻𝑋))
18		moni.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋))
19	18	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 ∈ (𝑍𝐻𝑋))
20	19, 16	eleqtrrd 2704	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 ∈ (𝑧𝐻𝑋))
21	20	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → 𝐾 ∈ (𝑧𝐻𝑋))
22		simpllr 799	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → 𝑧 = 𝑍)
23	22	opeq1d 4408	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → ⟨𝑧, 𝑋⟩ = ⟨𝑍, 𝑋⟩)
24	23	oveq1d 6665	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (⟨𝑧, 𝑋⟩ · 𝑌) = (⟨𝑍, 𝑋⟩ · 𝑌))
25		eqidd 2623	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → 𝐹 = 𝐹)
26		simplr 792	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → 𝑔 = 𝐺)
27	24, 25, 26	oveq123d 6671	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺))
28		simpr 477	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → ℎ = 𝐾)
29	24, 25, 28	oveq123d 6671	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾))
30	27, 29	eqeq12d 2637	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → ((𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) ↔ (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾)))
31	26, 28	eqeq12d 2637	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝑔 = ℎ ↔ 𝐺 = 𝐾))
32	30, 31	imbi12d 334	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (((𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) → 𝑔 = ℎ) ↔ ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) → 𝐺 = 𝐾)))
33	21, 32	rspcdv 3312	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → (∀ℎ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) → 𝑔 = ℎ) → ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) → 𝐺 = 𝐾)))
34	17, 33	rspcimdv 3310	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) → 𝑔 = ℎ) → ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) → 𝐺 = 𝐾)))
35	12, 34	rspcimdv 3310	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑧𝐻𝑋)∀ℎ ∈ (𝑧𝐻𝑋)((𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) → 𝑔 = ℎ) → ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) → 𝐺 = 𝐾)))
36	11, 35	mpd 15	. 2 ⊢ (𝜑 → ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) → 𝐺 = 𝐾))
37		oveq2 6658	. 2 ⊢ (𝐺 = 𝐾 → (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾))
38	36, 37	impbid1 215	1 ⊢ (𝜑 → ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) ↔ 𝐺 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ⟨cop 4183  ‘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-cat 16329  df-mon 16390

This theorem is referenced by:  epii  16403  monsect  16443  fthmon  16587  setcmon  16737