Theorem dfsmo2 7978

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 7977	. 2 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
2		ralcom 3354	. . . . . 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 453	. . . . . . . . 9 ⊢ (((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ (𝑦 ∈ dom 𝐹 → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
4		simpr 487	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
5		ordtr1 6229	. . . . . . . . . . . . . . 15 ⊢ (Ord dom 𝐹 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹))
6	5	3impib 1112	. . . . . . . . . . . . . 14 ⊢ ((Ord dom 𝐹 ∧ 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
7	6	3com23 1122	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
8		simp3 1134	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9	7, 8	jca 514	. . . . . . . . . . . 12 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥))
10	9	3expia 1117	. . . . . . . . . . 11 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝑦 ∈ 𝑥 → (𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥)))
11	4, 10	impbid2 228	. . . . . . . . . 10 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) ↔ 𝑦 ∈ 𝑥))
12	11	imbi1d 344	. . . . . . . . 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	syl5bbr 287	. . . . . . . 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 3195	. . . . . . 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 3196	. . . . . 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	syl5bb 285	. . . . 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 577	. . . 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 624	. . 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 1091	. . 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 1091	. . 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 305	. 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 277	1 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   ∈ wcel 2110  ∀wral 3138  dom cdm 5550  Ord word 6185  Oncon0 6186  ⟶wf 6346  ‘cfv 6350  Smo wsmo 7976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3497  df-in 3943  df-ss 3952  df-uni 4833  df-tr 5166  df-ord 6189  df-smo 7977

This theorem is referenced by:  issmo2  7980  smores2  7985  smofvon2  7987