Theorem dfsmo2 6403

Description: Alternate definition of a strictly monotone ordinal function. (Contributed by Mario Carneiro, 4-Mar-2013.)

Assertion

Ref	Expression
dfsmo2	⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))

Distinct variable group:   𝑥,𝐹,𝑦

Proof of Theorem dfsmo2

Step	Hyp	Ref	Expression
1		df-smo 6402	. 2 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
2		ralcom 2674	. . . . . 6 ⊢ (∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
3		impexp 263	. . . . . . . . 9 ⊢ (((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ (𝑦 ∈ dom 𝐹 → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
4		simpr 110	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
5		ordtr1 4456	. . . . . . . . . . . . . . 15 ⊢ (Ord dom 𝐹 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹))
6	5	3impib 1206	. . . . . . . . . . . . . 14 ⊢ ((Ord dom 𝐹 ∧ 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
7	6	3com23 1214	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
8		simp3 1004	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9	7, 8	jca 306	. . . . . . . . . . . 12 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥))
10	9	3expia 1210	. . . . . . . . . . 11 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝑦 ∈ 𝑥 → (𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥)))
11	4, 10	impbid2 143	. . . . . . . . . 10 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) ↔ 𝑦 ∈ 𝑥))
12	11	imbi1d 231	. . . . . . . . 9 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
13	3, 12	bitr3id 194	. . . . . . . 8 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹 → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))) ↔ (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
14	13	ralbidv2 2512	. . . . . . 7 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (∀𝑦 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
15	14	ralbidva 2506	. . . . . 6 ⊢ (Ord dom 𝐹 → (∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
16	2, 15	bitrid 192	. . . . 5 ⊢ (Ord dom 𝐹 → (∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
17	16	pm5.32i 454	. . . 4 ⊢ ((Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))) ↔ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
18	17	anbi2i 457	. . 3 ⊢ ((𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)))) ↔ (𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
19		3anass 987	. . 3 ⊢ ((𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))) ↔ (𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥)))))
20		3anass 987	. . 3 ⊢ ((𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ (𝐹:dom 𝐹⟶On ∧ (Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
21	18, 19, 20	3bitr4i 212	. 2 ⊢ ((𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))) ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))
22	1, 21	bitri 184	1 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 983   ∈ wcel 2180  ∀wral 2488  Ord word 4430  Oncon0 4431  dom cdm 4696  ⟶wf 5290  ‘cfv 5294  Smo wsmo 6401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-v 2781  df-in 3183  df-ss 3190  df-uni 3868  df-tr 4162  df-iord 4434  df-smo 6402

This theorem is referenced by:  issmo2  6405  smores2  6410  smofvon2dm  6412