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Theorem moni 17712

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 17710	. . . . 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 231	. . . 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 495	. . 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 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑍𝐻𝑋))
15		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝑧 = 𝑍)
16	15	oveq1d 7429	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝑧𝐻𝑋) = (𝑍𝐻𝑋))
17	14, 16	eleqtrrd 2832	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐺 ∈ (𝑧𝐻𝑋))
18		moni.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (𝑍𝐻𝑋))
19	18	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 ∈ (𝑍𝐻𝑋))
20	19, 16	eleqtrrd 2832	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → 𝐾 ∈ (𝑧𝐻𝑋))
21	20	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) → 𝐾 ∈ (𝑧𝐻𝑋))
22		simpllr 775	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → 𝑧 = 𝑍)
23	22	opeq1d 4875	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → ⟨𝑧, 𝑋⟩ = ⟨𝑍, 𝑋⟩)
24	23	oveq1d 7429	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (⟨𝑧, 𝑋⟩ · 𝑌) = (⟨𝑍, 𝑋⟩ · 𝑌))
25		eqidd 2729	. . . . . . . . 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 7435	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)𝑔) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺))
28		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → ℎ = 𝐾)
29	24, 25, 28	oveq123d 7435	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝐹(⟨𝑧, 𝑋⟩ · 𝑌)ℎ) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾))
30	27, 29	eqeq12d 2744	. . . . . . 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 2744	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑧 = 𝑍) ∧ 𝑔 = 𝐺) ∧ ℎ = 𝐾) → (𝑔 = ℎ ↔ 𝐺 = 𝐾))
32	30, 31	imbi12d 344	. . . . . 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 3600	. . . . 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 3598	. . . 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 3598	. . 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 7422	. 2 ⊢ (𝐺 = 𝐾 → (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾))
38	36, 37	impbid1 224	1 ⊢ (𝜑 → ((𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐺) = (𝐹(⟨𝑍, 𝑋⟩ · 𝑌)𝐾) ↔ 𝐺 = 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ⟨cop 4630 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 Hom chom 17237 compcco 17238 Catccat 17637 Monocmon 17704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-cat 17641 df-mon 17706

This theorem is referenced by: epii 17719 monsect 17759 fthmon 17909 setcmon 18069

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