Theorem dfsmo2 6184

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 6183	. 2 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹∀𝑥 ∈ dom 𝐹(𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
2		ralcom 2594	. . . . . 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 261	. . . . . . . . 9 ⊢ (((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (𝐹‘𝑥)) ↔ (𝑦 ∈ dom 𝐹 → (𝑦 ∈ 𝑥 → (𝐹‘𝑦) ∈ (𝐹‘𝑥))))
4		simpr 109	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
5		ordtr1 4310	. . . . . . . . . . . . . . 15 ⊢ (Ord dom 𝐹 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹))
6	5	3impib 1179	. . . . . . . . . . . . . 14 ⊢ ((Ord dom 𝐹 ∧ 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ dom 𝐹) → 𝑦 ∈ dom 𝐹)
7	6	3com23 1187	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
8		simp3 983	. . . . . . . . . . . . 13 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
9	7, 8	jca 304	. . . . . . . . . . . 12 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥))
10	9	3expia 1183	. . . . . . . . . . 11 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝑦 ∈ 𝑥 → (𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥)))
11	4, 10	impbid2 142	. . . . . . . . . 10 ⊢ ((Ord dom 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝑦 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝑥) ↔ 𝑦 ∈ 𝑥))
12	11	imbi1d 230	. . . . . . . . 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 193	. . . . . . . 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 2439	. . . . . . 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 2433	. . . . . 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 191	. . . . 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 449	. . . 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 452	. . 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 966	. . 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 966	. . 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 211	. 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 183	1 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   ∈ wcel 1480  ∀wral 2416  Ord word 4284  Oncon0 4285  dom cdm 4539  ⟶wf 5119  ‘cfv 5123  Smo wsmo 6182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-uni 3737  df-tr 4027  df-iord 4288  df-smo 6183

This theorem is referenced by:  issmo2  6186  smores2  6191  smofvon2dm  6193