Theorem dfsmo2 6278

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 6277	. 2 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
2		ralcom 2638	. . . . . 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 4382	. . . . . . . . . . . . . . 15 ⊢ (Ord dom 𝐹 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹))
6	5	3impib 1201	. . . . . . . . . . . . . 14 ⊢ ((Ord dom 𝐹 ∧ 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
7	6	3com23 1209	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
8		simp3 999	. . . . . . . . . . . . 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 1205	. . . . . . . . . . 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 2477	. . . . . . 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 2471	. . . . . 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 982	. . 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 982	. . 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 978   ∈ wcel 2146  ∀wral 2453  Ord word 4356  Oncon0 4357  dom cdm 4620  ⟶wf 5204  ‘cfv 5208  Smo wsmo 6276

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-in 3133  df-ss 3140  df-uni 3806  df-tr 4097  df-iord 4360  df-smo 6277

This theorem is referenced by:  issmo2  6280  smores2  6285  smofvon2dm  6287