Theorem moni 17070

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 17068	. . . . 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 235	. . . 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 499	. . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑍𝐻𝑋))
15		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍)
16	15	oveq1d 7170	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝑧𝐻𝑋) = (𝑍𝐻𝑋))
17	14, 16	eleqtrrd 2855	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑧𝐻𝑋))
18		moni.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋))
19	18	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 ∈ (𝑍𝐻𝑋))
20	19, 16	eleqtrrd 2855	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 ∈ (𝑧𝐻𝑋))
21	20	adantr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → 𝐾 ∈ (𝑧𝐻𝑋))
22		simpllr 775	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → 𝑧 = 𝑍)
23	22	opeq1d 4772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → ⟨𝑧, 𝑋⟩ = ⟨𝑍, 𝑋⟩)
24	23	oveq1d 7170	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (⟨𝑧, 𝑋⟩ · 𝑌) = (⟨𝑍, 𝑋⟩ · 𝑌))
25		eqidd 2759	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → 𝐹 = 𝐹)
26		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → 𝑔 = 𝐺)
27	24, 25, 26	oveq123d 7176	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺))
28		simpr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → ℎ = 𝐾)
29	24, 25, 28	oveq123d 7176	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾))
30	27, 29	eqeq12d 2774	. . . . . . 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 2774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝑔 = ℎ ↔ 𝐺 = 𝐾))
32	30, 31	imbi12d 348	. . . . . 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 3535	. . . . 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 3533	. . . 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 3533	. . 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 7163	. 2 ⊢ (𝐺 = 𝐾 → (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾))
38	36, 37	impbid1 228	1 ⊢ (𝜑 → ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) ↔ 𝐺 = 𝐾))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ⟨cop 4531  ‘cfv 6339  (class class class)co 7155  Basecbs 16546  Hom chom 16639  compcco 16640  Catccat 16998  Monocmon 17062

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7698  df-2nd 7699  df-cat 17002  df-mon 17064

This theorem is referenced by:  epii  17077  monsect  17117  fthmon  17261  setcmon  17418