Theorem smo11 7461

Description: A strictly monotone ordinal function is one-to-one. (Contributed by Mario Carneiro, 28-Feb-2013.)

Assertion

Ref	Expression
smo11	⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–1-1→𝐵)

Proof of Theorem smo11

Dummy variables 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹) → 𝐹:𝐴⟶𝐵)
2		ffn 6045	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		smodm2 7452	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
4		ordelord 5745	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑧 ∈ 𝐴) → Ord 𝑧)
5	4	ex 450	. . . . . . 7 ⊢ (Ord 𝐴 → (𝑧 ∈ 𝐴 → Ord 𝑧))
6	3, 5	syl 17	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑧 ∈ 𝐴 → Ord 𝑧))
7		ordelord 5745	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑤 ∈ 𝐴) → Ord 𝑤)
8	7	ex 450	. . . . . . 7 ⊢ (Ord 𝐴 → (𝑤 ∈ 𝐴 → Ord 𝑤))
9	3, 8	syl 17	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑤 ∈ 𝐴 → Ord 𝑤))
10	6, 9	anim12d 586	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (Ord 𝑧 ∧ Ord 𝑤)))
11		ordtri3or 5755	. . . . . . 7 ⊢ ((Ord 𝑧 ∧ Ord 𝑤) → (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧))
12		simp1rr 1127	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑤 ∈ 𝐴)
13		smoel2 7460	. . . . . . . . . . . . . 14 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝐹‘𝑦) ∈ (𝐹‘𝑥))
14	13	ralrimivva 2971	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
15	14	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
16	15	3ad2ant1 1082	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
17		simp2 1062	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑧 ∈ 𝑤)
18		simp3 1063	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑧) = (𝐹‘𝑤))
19		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
20	19	eleq2d 2687	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑤)))
21	20	raleqbi1dv 3146	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝑤 (𝐹‘𝑦) ∈ (𝐹‘𝑤)))
22	21	rspcv 3305	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) → ∀𝑦 ∈ 𝑤 (𝐹‘𝑦) ∈ (𝐹‘𝑤)))
23		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
24	23	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) ∈ (𝐹‘𝑤) ↔ (𝐹‘𝑧) ∈ (𝐹‘𝑤)))
25	24	rspccv 3306	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑤 (𝐹‘𝑦) ∈ (𝐹‘𝑤) → (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤)))
26	22, 25	syl6 35	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) → (𝑧 ∈ 𝑤 → (𝐹‘𝑧) ∈ (𝐹‘𝑤))))
27	26	3imp 1256	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ∧ 𝑧 ∈ 𝑤) → (𝐹‘𝑧) ∈ (𝐹‘𝑤))
28		eleq1 2689	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) = (𝐹‘𝑤) → ((𝐹‘𝑧) ∈ (𝐹‘𝑤) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑤)))
29	28	biimpac 503	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ (𝐹‘𝑤) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑤) ∈ (𝐹‘𝑤))
30	27, 29	sylan 488	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ∧ 𝑧 ∈ 𝑤) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑤) ∈ (𝐹‘𝑤))
31	12, 16, 17, 18, 30	syl31anc 1329	. . . . . . . . . 10 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑤) ∈ (𝐹‘𝑤))
32		smofvon2 7453	. . . . . . . . . . . . 13 ⊢ (Smo 𝐹 → (𝐹‘𝑤) ∈ On)
33		eloni 5733	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑤) ∈ On → Ord (𝐹‘𝑤))
34		ordirr 5741	. . . . . . . . . . . . 13 ⊢ (Ord (𝐹‘𝑤) → ¬ (𝐹‘𝑤) ∈ (𝐹‘𝑤))
35	32, 33, 34	3syl 18	. . . . . . . . . . . 12 ⊢ (Smo 𝐹 → ¬ (𝐹‘𝑤) ∈ (𝐹‘𝑤))
36	35	ad2antlr 763	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ¬ (𝐹‘𝑤) ∈ (𝐹‘𝑤))
37	36	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → ¬ (𝐹‘𝑤) ∈ (𝐹‘𝑤))
38	31, 37	pm2.21dd 186	. . . . . . . . 9 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑧 = 𝑤)
39	38	3exp 1264	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧 ∈ 𝑤 → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
40		ax-1 6	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
41	40	a1i 11	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧 = 𝑤 → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
42		simp1rl 1126	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑧 ∈ 𝐴)
43	15	3ad2ant1 1082	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))
44		simp2 1062	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑤 ∈ 𝑧)
45		simp3 1063	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑧) = (𝐹‘𝑤))
46		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
47	46	eleq2d 2687	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑧)))
48	47	raleqbi1dv 3146	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝑧 (𝐹‘𝑦) ∈ (𝐹‘𝑧)))
49	48	rspcv 3305	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) → ∀𝑦 ∈ 𝑧 (𝐹‘𝑦) ∈ (𝐹‘𝑧)))
50		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
51	50	eleq1d 2686	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧)))
52	51	rspccv 3306	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑧 (𝐹‘𝑦) ∈ (𝐹‘𝑧) → (𝑤 ∈ 𝑧 → (𝐹‘𝑤) ∈ (𝐹‘𝑧)))
53	49, 52	syl6 35	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) → (𝑤 ∈ 𝑧 → (𝐹‘𝑤) ∈ (𝐹‘𝑧))))
54	53	3imp 1256	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) → (𝐹‘𝑤) ∈ (𝐹‘𝑧))
55		eleq2 2690	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) = (𝐹‘𝑤) → ((𝐹‘𝑤) ∈ (𝐹‘𝑧) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑤)))
56	55	biimpac 503	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑤) ∈ (𝐹‘𝑧) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑤) ∈ (𝐹‘𝑤))
57	54, 56	sylan 488	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑤) ∈ (𝐹‘𝑤))
58	42, 43, 44, 45, 57	syl31anc 1329	. . . . . . . . . 10 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑤) ∈ (𝐹‘𝑤))
59	36	3ad2ant1 1082	. . . . . . . . . 10 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → ¬ (𝐹‘𝑤) ∈ (𝐹‘𝑤))
60	58, 59	pm2.21dd 186	. . . . . . . . 9 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑧 = 𝑤)
61	60	3exp 1264	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑤 ∈ 𝑧 → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
62	39, 41, 61	3jaod 1392	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
63	11, 62	syl5 34	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((Ord 𝑧 ∧ Ord 𝑤) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
64	63	ex 450	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → ((Ord 𝑧 ∧ Ord 𝑤) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))))
65	10, 64	mpdd 43	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
66	65	ralrimivv 2970	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
67	2, 66	sylan 488	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹) → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
68		dff13 6512	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
69	1, 67, 68	sylanbrc 698	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–1-1→𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Ord word 5722  Oncon0 5723   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888  Smo wsmo 7442

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896  df-smo 7443

This theorem is referenced by:  smoiso2  7466  alephf1ALT  8926