Theorem moni 17831

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 17829	. . . . 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 501	. . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑍𝐻𝑋))
15		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍)
16	15	oveq1d 7432	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝑧𝐻𝑋) = (𝑍𝐻𝑋))
17	14, 16	eleqtrrd 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑧𝐻𝑋))
18		moni.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋))
19	18	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 ∈ (𝑍𝐻𝑋))
20	19, 16	eleqtrrd 2865	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 ∈ (𝑧𝐻𝑋))
21	20	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → 𝐾 ∈ (𝑧𝐻𝑋))
22		simpllr 788	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → 𝑧 = 𝑍)
23	22	opeq1d 4842	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → ⟨𝑧, 𝑋⟩ = ⟨𝑍, 𝑋⟩)
24	23	oveq1d 7432	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (⟨𝑧, 𝑋⟩ · 𝑌) = (⟨𝑍, 𝑋⟩ · 𝑌))
25		eqidd 2763	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → 𝐹 = 𝐹)
26		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → 𝑔 = 𝐺)
27	24, 25, 26	oveq123d 7438	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺))
28		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → ℎ = 𝐾)
29	24, 25, 28	oveq123d 7438	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾))
30	27, 29	eqeq12d 2778	. . . . . . 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 2778	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝑔 = ℎ ↔ 𝐺 = 𝐾))
32	30, 31	imbi12d 347	. . . . . 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 3571	. . . . 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 3569	. . . 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 3569	. . 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 16	. 2 ⊢ (𝜑 → ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) → 𝐺 = 𝐾))
37		oveq2 7425	. 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ⟨cop 4593  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  Hom chom 17359  compcco 17360  Catccat 17758  Monocmon 17823

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-cat 17762  df-mon 17825

This theorem is used by:  epii  17838  monsect  17878  fthmon  18024  setcmon  18182