Theorem smo11 8195

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 483	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹) → 𝐹:𝐴⟶𝐵)
2		ffn 6600	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		smodm2 8186	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
4		ordelord 6288	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑧 ∈ 𝐴) → Ord 𝑧)
5	4	ex 413	. . . . . . 7 ⊢ (Ord 𝐴 → (𝑧 ∈ 𝐴 → Ord 𝑧))
6	3, 5	syl 17	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑧 ∈ 𝐴 → Ord 𝑧))
7		ordelord 6288	. . . . . . . 8 ⊢ ((Ord 𝐴 ∧ 𝑤 ∈ 𝐴) → Ord 𝑤)
8	7	ex 413	. . . . . . 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 609	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (Ord 𝑧 ∧ Ord 𝑤)))
11		ordtri3or 6298	. . . . . . 7 ⊢ ((Ord 𝑧 ∧ Ord 𝑤) → (𝑧 ∈ 𝑤 ∨ 𝑧 = 𝑤 ∨ 𝑤 ∈ 𝑧))
12		simp1rr 1238	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑤 ∈ 𝐴)
13		smoel2 8194	. . . . . . . . . . . . . 14 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝐹‘𝑦) ∈ (𝐹‘𝑥))
14	13	ralrimivva 3123	. . . . . . . . . . . . 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 1132	. . . . . . . . . . 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 1136	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑧 ∈ 𝑤)
18		simp3 1137	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑧) = (𝐹‘𝑤))
19		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
20	19	eleq2d 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑤)))
21	20	raleqbi1dv 3340	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑤 → (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝑤 (𝐹‘𝑦) ∈ (𝐹‘𝑤)))
22	21	rspcv 3557	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) → ∀𝑦 ∈ 𝑤 (𝐹‘𝑦) ∈ (𝐹‘𝑤)))
23		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
24	23	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) ∈ (𝐹‘𝑤) ↔ (𝐹‘𝑧) ∈ (𝐹‘𝑤)))
25	24	rspccv 3558	. . . . . . . . . . . . . 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 1110	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ∧ 𝑧 ∈ 𝑤) → (𝐹‘𝑧) ∈ (𝐹‘𝑤))
28		eleq1 2826	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) = (𝐹‘𝑤) → ((𝐹‘𝑧) ∈ (𝐹‘𝑤) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑤)))
29	28	biimpac 479	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑧) ∈ (𝐹‘𝑤) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑤) ∈ (𝐹‘𝑤))
30	27, 29	sylan 580	. . . . . . . . . . 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 1372	. . . . . . . . . 10 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑤) ∈ (𝐹‘𝑤))
32		smofvon2 8187	. . . . . . . . . . . . 13 ⊢ (Smo 𝐹 → (𝐹‘𝑤) ∈ On)
33		eloni 6276	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑤) ∈ On → Ord (𝐹‘𝑤))
34		ordirr 6284	. . . . . . . . . . . . 13 ⊢ (Ord (𝐹‘𝑤) → ¬ (𝐹‘𝑤) ∈ (𝐹‘𝑤))
35	32, 33, 34	3syl 18	. . . . . . . . . . . 12 ⊢ (Smo 𝐹 → ¬ (𝐹‘𝑤) ∈ (𝐹‘𝑤))
36	35	ad2antlr 724	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ¬ (𝐹‘𝑤) ∈ (𝐹‘𝑤))
37	36	3ad2ant1 1132	. . . . . . . . . 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 194	. . . . . . . . 9 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑧 ∈ 𝑤 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑧 = 𝑤)
39	38	3exp 1118	. . . . . . . 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 1237	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑧 ∈ 𝐴)
43	15	3ad2ant1 1132	. . . . . . . . . . 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 1136	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑤 ∈ 𝑧)
45		simp3 1137	. . . . . . . . . . 11 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑧) = (𝐹‘𝑤))
46		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
47	46	eleq2d 2824	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ∈ (𝐹‘𝑧)))
48	47	raleqbi1dv 3340	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝑧 (𝐹‘𝑦) ∈ (𝐹‘𝑧)))
49	48	rspcv 3557	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) → ∀𝑦 ∈ 𝑧 (𝐹‘𝑦) ∈ (𝐹‘𝑧)))
50		fveq2 6774	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
51	50	eleq1d 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑦) ∈ (𝐹‘𝑧) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑧)))
52	51	rspccv 3558	. . . . . . . . . . . . . 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 1110	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ 𝑧) → (𝐹‘𝑤) ∈ (𝐹‘𝑧))
55		eleq2 2827	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑧) = (𝐹‘𝑤) → ((𝐹‘𝑤) ∈ (𝐹‘𝑧) ↔ (𝐹‘𝑤) ∈ (𝐹‘𝑤)))
56	55	biimpac 479	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑤) ∈ (𝐹‘𝑧) ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑤) ∈ (𝐹‘𝑤))
57	54, 56	sylan 580	. . . . . . . . . . 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 1372	. . . . . . . . . 10 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → (𝐹‘𝑤) ∈ (𝐹‘𝑤))
59	36	3ad2ant1 1132	. . . . . . . . . 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 194	. . . . . . . . 9 ⊢ ((((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ 𝑤 ∈ 𝑧 ∧ (𝐹‘𝑧) = (𝐹‘𝑤)) → 𝑧 = 𝑤)
61	60	3exp 1118	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑤 ∈ 𝑧 → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
62	39, 41, 61	3jaod 1427	. . . . . . 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 413	. . . . 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 3122	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
67	2, 66	sylan 580	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹) → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
68		dff13 7128	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
69	1, 67, 68	sylanbrc 583	1 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Smo 𝐹) → 𝐹:𝐴–1-1→𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ w3o 1085   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Ord word 6265  Oncon0 6266   Fn wfn 6428  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433  Smo wsmo 8176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-ord 6269  df-on 6270  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441  df-smo 8177

This theorem is referenced by:  smoiso2  8200  alephf1ALT  9859