Theorem icmpmon 10587

Description: If (GRF) is monic then F is monic. JFM CAT₁ th. 62

Hypotheses

Ref	Expression
icmpmon.1	⊢ O = dom (id ‘T)
icmpmon.2	⊢ H = (hom ‘T)
icmpmon.3	⊢ R = (o ‘T)

Assertion

Ref	Expression
icmpmon	⊢ ((T ∈ Cat ⋀ ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) ⋀ (GRF) ∈ (Monic ‘T))) → F ∈ (Monic ‘T))

Proof of Theorem icmpmon

Step	Hyp	Ref	Expression
1		icmpmon.1	. . . . . . 7 ⊢ O = dom (id ‘T)
2		icmpmon.2	. . . . . . 7 ⊢ H = (hom ‘T)
3		icmpmon.3	. . . . . . 7 ⊢ R = (o ‘T)
4	1, 2, 3	ismonc 10584	. . . . . 6 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ C ∈ O) ⋀ (GRF) ∈ (H ‘⟨A, C⟩)) → ((GRF) ∈ (Monic ‘T) ↔ ∀a ∈ O ∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)(((GRF)Rm) = ((GRF)Rn) → m = n)))
5		3simp1 786	. . . . . 6 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → T ∈ Cat)
6		3simpb 784	. . . . . . 7 ⊢ ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) → (A ∈ O ⋀ C ∈ O))
7	6	3ad2ant2 799	. . . . . 6 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → (A ∈ O ⋀ C ∈ O))
8	1, 2, 3	homgrf 10574	. . . . . . 7 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O)) → ((F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (GRF) ∈ (H ‘⟨A, C⟩)))
9	8	3impia 828	. . . . . 6 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → (GRF) ∈ (H ‘⟨A, C⟩))
10	4, 5, 7, 9	syl3anc 856	. . . . 5 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → ((GRF) ∈ (Monic ‘T) ↔ ∀a ∈ O ∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)(((GRF)Rm) = ((GRF)Rn) → m = n)))
11		pm3.27 323	. . . . . . . . . . . . 13 ⊢ (((((GRF)Rm) = (GR(FRm)) ⋀ ((GRF)Rn) = (GR(FRn))) ⋀ (GR(FRm)) = (GR(FRn))) → (GR(FRm)) = (GR(FRn)))
12		simpll 412	. . . . . . . . . . . . 13 ⊢ (((((GRF)Rm) = (GR(FRm)) ⋀ ((GRF)Rn) = (GR(FRn))) ⋀ (GR(FRm)) = (GR(FRn))) → ((GRF)Rm) = (GR(FRm)))
13		simplr 413	. . . . . . . . . . . . 13 ⊢ (((((GRF)Rm) = (GR(FRm)) ⋀ ((GRF)Rn) = (GR(FRn))) ⋀ (GR(FRm)) = (GR(FRn))) → ((GRF)Rn) = (GR(FRn)))
14	11, 12, 13	3eqtr4d 1509	. . . . . . . . . . . 12 ⊢ (((((GRF)Rm) = (GR(FRm)) ⋀ ((GRF)Rn) = (GR(FRn))) ⋀ (GR(FRm)) = (GR(FRn))) → ((GRF)Rm) = ((GRF)Rn))
15	14	ex 373	. . . . . . . . . . 11 ⊢ ((((GRF)Rm) = (GR(FRm)) ⋀ ((GRF)Rn) = (GR(FRn))) → ((GR(FRm)) = (GR(FRn)) → ((GRF)Rm) = ((GRF)Rn)))
16	15	imim1d 28	. . . . . . . . . 10 ⊢ ((((GRF)Rm) = (GR(FRm)) ⋀ ((GRF)Rn) = (GR(FRn))) → ((((GRF)Rm) = ((GRF)Rn) → m = n) → ((GR(FRm)) = (GR(FRn)) → m = n)))
17		opreq2 3954	. . . . . . . . . 10 ⊢ ((FRm) = (FRn) → (GR(FRm)) = (GR(FRn)))
18	16, 17	syl7 23	. . . . . . . . 9 ⊢ ((((GRF)Rm) = (GR(FRm)) ⋀ ((GRF)Rn) = (GR(FRn))) → ((((GRF)Rm) = ((GRF)Rn) → m = n) → ((FRm) = (FRn) → m = n)))
19	1, 2, 3	cmpassoh 10573	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((T ∈ Cat ⋀ (a ∈ O ⋀ A ∈ O) ⋀ (B ∈ O ⋀ C ∈ O)) → ((m ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (GR(FRm)) = ((GRF)Rm)))
20	19	3exp 830	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (T ∈ Cat → ((a ∈ O ⋀ A ∈ O) → ((B ∈ O ⋀ C ∈ O) → ((m ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (GR(FRm)) = ((GRF)Rm)))))
21	20	com3l 34	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((a ∈ O ⋀ A ∈ O) → ((B ∈ O ⋀ C ∈ O) → (T ∈ Cat → ((m ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (GR(FRm)) = ((GRF)Rm)))))
22	21	exp4b 379	. . . . . . . . . . . . . . . . . . 19 ⊢ (a ∈ O → (A ∈ O → (B ∈ O → (C ∈ O → (T ∈ Cat → ((m ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (GR(FRm)) = ((GRF)Rm)))))))
23	22	3impd 845	. . . . . . . . . . . . . . . . . 18 ⊢ (a ∈ O → ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) → (T ∈ Cat → ((m ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (GR(FRm)) = ((GRF)Rm)))))
24	23	com13 33	. . . . . . . . . . . . . . . . 17 ⊢ (T ∈ Cat → ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) → (a ∈ O → ((m ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (GR(FRm)) = ((GRF)Rm)))))
25	24	imp 350	. . . . . . . . . . . . . . . 16 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O)) → (a ∈ O → ((m ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (GR(FRm)) = ((GRF)Rm))))
26	25	com13 33	. . . . . . . . . . . . . . 15 ⊢ ((m ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (a ∈ O → ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O)) → (GR(FRm)) = ((GRF)Rm))))
27	26	3expib 834	. . . . . . . . . . . . . 14 ⊢ (m ∈ (H ‘⟨a, A⟩) → ((F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (a ∈ O → ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O)) → (GR(FRm)) = ((GRF)Rm)))))
28	27	com14 38	. . . . . . . . . . . . 13 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O)) → ((F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (a ∈ O → (m ∈ (H ‘⟨a, A⟩) → (GR(FRm)) = ((GRF)Rm)))))
29	28	3impia 828	. . . . . . . . . . . 12 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → (a ∈ O → (m ∈ (H ‘⟨a, A⟩) → (GR(FRm)) = ((GRF)Rm))))
30	29	imp31 362	. . . . . . . . . . 11 ⊢ ((((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) ⋀ a ∈ O) ⋀ m ∈ (H ‘⟨a, A⟩)) → (GR(FRm)) = ((GRF)Rm))
31	30	eqcomd 1472	. . . . . . . . . 10 ⊢ ((((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) ⋀ a ∈ O) ⋀ m ∈ (H ‘⟨a, A⟩)) → ((GRF)Rm) = (GR(FRm)))
32	31	adantr 389	. . . . . . . . 9 ⊢ (((((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) ⋀ a ∈ O) ⋀ m ∈ (H ‘⟨a, A⟩)) ⋀ n ∈ (H ‘⟨a, A⟩)) → ((GRF)Rm) = (GR(FRm)))
33	1, 2, 3	cmpassoh 10573	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((T ∈ Cat ⋀ (a ∈ O ⋀ A ∈ O) ⋀ (B ∈ O ⋀ C ∈ O)) → ((n ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (GR(FRn)) = ((GRF)Rn)))
34	33	imp 350	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((T ∈ Cat ⋀ (a ∈ O ⋀ A ∈ O) ⋀ (B ∈ O ⋀ C ∈ O)) ⋀ (n ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → (GR(FRn)) = ((GRF)Rn))
35	34	eqcomd 1472	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((T ∈ Cat ⋀ (a ∈ O ⋀ A ∈ O) ⋀ (B ∈ O ⋀ C ∈ O)) ⋀ (n ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → ((GRF)Rn) = (GR(FRn)))
36	35	3exp1 847	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (T ∈ Cat → ((a ∈ O ⋀ A ∈ O) → ((B ∈ O ⋀ C ∈ O) → ((n ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → ((GRF)Rn) = (GR(FRn))))))
37	36	com3l 34	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((a ∈ O ⋀ A ∈ O) → ((B ∈ O ⋀ C ∈ O) → (T ∈ Cat → ((n ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → ((GRF)Rn) = (GR(FRn))))))
38	37	exp4b 379	. . . . . . . . . . . . . . . . . . 19 ⊢ (a ∈ O → (A ∈ O → (B ∈ O → (C ∈ O → (T ∈ Cat → ((n ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → ((GRF)Rn) = (GR(FRn))))))))
39	38	3impd 845	. . . . . . . . . . . . . . . . . 18 ⊢ (a ∈ O → ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) → (T ∈ Cat → ((n ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → ((GRF)Rn) = (GR(FRn))))))
40	39	com14 38	. . . . . . . . . . . . . . . . 17 ⊢ ((n ∈ (H ‘⟨a, A⟩) ⋀ F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) → (T ∈ Cat → (a ∈ O → ((GRF)Rn) = (GR(FRn))))))
41	40	3expib 834	. . . . . . . . . . . . . . . 16 ⊢ (n ∈ (H ‘⟨a, A⟩) → ((F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) → (T ∈ Cat → (a ∈ O → ((GRF)Rn) = (GR(FRn)))))))
42	41	com23 32	. . . . . . . . . . . . . . 15 ⊢ (n ∈ (H ‘⟨a, A⟩) → ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) → ((F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (T ∈ Cat → (a ∈ O → ((GRF)Rn) = (GR(FRn)))))))
43	42	com14 38	. . . . . . . . . . . . . 14 ⊢ (T ∈ Cat → ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) → ((F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → (n ∈ (H ‘⟨a, A⟩) → (a ∈ O → ((GRF)Rn) = (GR(FRn)))))))
44	43	3imp 825	. . . . . . . . . . . . 13 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → (n ∈ (H ‘⟨a, A⟩) → (a ∈ O → ((GRF)Rn) = (GR(FRn)))))
45	44	com23 32	. . . . . . . . . . . 12 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → (a ∈ O → (n ∈ (H ‘⟨a, A⟩) → ((GRF)Rn) = (GR(FRn)))))
46	45	imp 350	. . . . . . . . . . 11 ⊢ (((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) ⋀ a ∈ O) → (n ∈ (H ‘⟨a, A⟩) → ((GRF)Rn) = (GR(FRn))))
47	46	adantr 389	. . . . . . . . . 10 ⊢ ((((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) ⋀ a ∈ O) ⋀ m ∈ (H ‘⟨a, A⟩)) → (n ∈ (H ‘⟨a, A⟩) → ((GRF)Rn) = (GR(FRn))))
48	47	imp 350	. . . . . . . . 9 ⊢ (((((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) ⋀ a ∈ O) ⋀ m ∈ (H ‘⟨a, A⟩)) ⋀ n ∈ (H ‘⟨a, A⟩)) → ((GRF)Rn) = (GR(FRn)))
49	18, 32, 48	sylanc 471	. . . . . . . 8 ⊢ (((((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) ⋀ a ∈ O) ⋀ m ∈ (H ‘⟨a, A⟩)) ⋀ n ∈ (H ‘⟨a, A⟩)) → ((((GRF)Rm) = ((GRF)Rn) → m = n) → ((FRm) = (FRn) → m = n)))
50	49	r19.20dva 1701	. . . . . . 7 ⊢ ((((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) ⋀ a ∈ O) ⋀ m ∈ (H ‘⟨a, A⟩)) → (∀n ∈ (H ‘⟨a, A⟩)(((GRF)Rm) = ((GRF)Rn) → m = n) → ∀n ∈ (H ‘⟨a, A⟩)((FRm) = (FRn) → m = n)))
51	50	r19.20dva 1701	. . . . . 6 ⊢ (((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) ⋀ a ∈ O) → (∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)(((GRF)Rm) = ((GRF)Rn) → m = n) → ∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)((FRm) = (FRn) → m = n)))
52	51	r19.20dva 1701	. . . . 5 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → (∀a ∈ O ∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)(((GRF)Rm) = ((GRF)Rn) → m = n) → ∀a ∈ O ∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)((FRm) = (FRn) → m = n)))
53	10, 52	sylbid 203	. . . 4 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩))) → ((GRF) ∈ (Monic ‘T) → ∀a ∈ O ∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)((FRm) = (FRn) → m = n)))
54	53	3exp 830	. . 3 ⊢ (T ∈ Cat → ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) → ((F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) → ((GRF) ∈ (Monic ‘T) → ∀a ∈ O ∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)((FRm) = (FRn) → m = n)))))
55	54	3imp2 846	. 2 ⊢ ((T ∈ Cat ⋀ ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) ⋀ (GRF) ∈ (Monic ‘T))) → ∀a ∈ O ∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)((FRm) = (FRn) → m = n))
56	1, 2, 3	ismonc 10584	. . 3 ⊢ ((T ∈ Cat ⋀ (A ∈ O ⋀ B ∈ O) ⋀ F ∈ (H ‘⟨A, B⟩)) → (F ∈ (Monic ‘T) ↔ ∀a ∈ O ∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)((FRm) = (FRn) → m = n)))
57		pm3.26 319	. . 3 ⊢ ((T ∈ Cat ⋀ ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) ⋀ (GRF) ∈ (Monic ‘T))) → T ∈ Cat)
58		3simpa 783	. . . . 5 ⊢ ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) → (A ∈ O ⋀ B ∈ O))
59	58	3ad2ant1 798	. . . 4 ⊢ (((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) ⋀ (GRF) ∈ (Monic ‘T)) → (A ∈ O ⋀ B ∈ O))
60	59	adantl 388	. . 3 ⊢ ((T ∈ Cat ⋀ ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) ⋀ (GRF) ∈ (Monic ‘T))) → (A ∈ O ⋀ B ∈ O))
61		3simp2 787	. . . . 5 ⊢ (((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) ⋀ (GRF) ∈ (Monic ‘T)) → (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)))
62	61	pm3.26d 321	. . . 4 ⊢ (((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) ⋀ (GRF) ∈ (Monic ‘T)) → F ∈ (H ‘⟨A, B⟩))
63	62	adantl 388	. . 3 ⊢ ((T ∈ Cat ⋀ ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) ⋀ (GRF) ∈ (Monic ‘T))) → F ∈ (H ‘⟨A, B⟩))
64	56, 57, 60, 63	syl3anc 856	. 2 ⊢ ((T ∈ Cat ⋀ ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) ⋀ (GRF) ∈ (Monic ‘T))) → (F ∈ (Monic ‘T) ↔ ∀a ∈ O ∀m ∈ (H ‘⟨a, A⟩)∀n ∈ (H ‘⟨a, A⟩)((FRm) = (FRn) → m = n)))
65	55, 64	mpbird 196	1 ⊢ ((T ∈ Cat ⋀ ((A ∈ O ⋀ B ∈ O ⋀ C ∈ O) ⋀ (F ∈ (H ‘⟨A, B⟩) ⋀ G ∈ (H ‘⟨B, C⟩)) ⋀ (GRF) ∈ (Monic ‘T))) → F ∈ (Monic ‘T))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   ⋀ w3a 773   = wceq 953   ∈ wcel 955  ∀wral 1637  ⟨cop 2401  dom cdm 3160   ‘cfv 3172  (class class class)co 3948  idcid_ 10490  oco_ 10491  Catccat 10529  homchom 10557  Moniccmon 10576

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-alg 10492  df-doma 10493  df-coda 10494  df-ida 10495  df-cmpa 10496  df-ded 10512  df-cat 10530  df-hom 10558  df-mon 10578

Copyright terms: Public domain