Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  smores2 Structured version   Visualization version   GIF version

Theorem smores2 7734
 Description: A strictly monotone ordinal function restricted to an ordinal is still monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)
Assertion
Ref Expression
smores2 ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹𝐴))

Proof of Theorem smores2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsmo2 7727 . . . . . . 7 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
21simp1bi 1136 . . . . . 6 (Smo 𝐹𝐹:dom 𝐹⟶On)
32ffund 6295 . . . . 5 (Smo 𝐹 → Fun 𝐹)
4 funres 6177 . . . . . 6 (Fun 𝐹 → Fun (𝐹𝐴))
54funfnd 6166 . . . . 5 (Fun 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
63, 5syl 17 . . . 4 (Smo 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
7 df-ima 5368 . . . . . 6 (𝐹𝐴) = ran (𝐹𝐴)
8 imassrn 5731 . . . . . 6 (𝐹𝐴) ⊆ ran 𝐹
97, 8eqsstr3i 3855 . . . . 5 ran (𝐹𝐴) ⊆ ran 𝐹
102frnd 6298 . . . . 5 (Smo 𝐹 → ran 𝐹 ⊆ On)
119, 10syl5ss 3832 . . . 4 (Smo 𝐹 → ran (𝐹𝐴) ⊆ On)
12 df-f 6139 . . . 4 ((𝐹𝐴):dom (𝐹𝐴)⟶On ↔ ((𝐹𝐴) Fn dom (𝐹𝐴) ∧ ran (𝐹𝐴) ⊆ On))
136, 11, 12sylanbrc 578 . . 3 (Smo 𝐹 → (𝐹𝐴):dom (𝐹𝐴)⟶On)
1413adantr 474 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → (𝐹𝐴):dom (𝐹𝐴)⟶On)
15 smodm 7731 . . 3 (Smo 𝐹 → Ord dom 𝐹)
16 ordin 6006 . . . . 5 ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord (𝐴 ∩ dom 𝐹))
17 dmres 5668 . . . . . 6 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
18 ordeq 5983 . . . . . 6 (dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹) → (Ord dom (𝐹𝐴) ↔ Ord (𝐴 ∩ dom 𝐹)))
1917, 18ax-mp 5 . . . . 5 (Ord dom (𝐹𝐴) ↔ Ord (𝐴 ∩ dom 𝐹))
2016, 19sylibr 226 . . . 4 ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord dom (𝐹𝐴))
2120ancoms 452 . . 3 ((Ord dom 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹𝐴))
2215, 21sylan 575 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹𝐴))
23 resss 5671 . . . . . 6 (𝐹𝐴) ⊆ 𝐹
24 dmss 5568 . . . . . 6 ((𝐹𝐴) ⊆ 𝐹 → dom (𝐹𝐴) ⊆ dom 𝐹)
2523, 24ax-mp 5 . . . . 5 dom (𝐹𝐴) ⊆ dom 𝐹
261simp3bi 1138 . . . . 5 (Smo 𝐹 → ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
27 ssralv 3885 . . . . 5 (dom (𝐹𝐴) ⊆ dom 𝐹 → (∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
2825, 26, 27mpsyl 68 . . . 4 (Smo 𝐹 → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
2928adantr 474 . . 3 ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
30 ordtr1 6019 . . . . . . . . . . 11 (Ord dom (𝐹𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦 ∈ dom (𝐹𝐴)))
3122, 30syl 17 . . . . . . . . . 10 ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦 ∈ dom (𝐹𝐴)))
32 inss1 4053 . . . . . . . . . . . 12 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
3317, 32eqsstri 3854 . . . . . . . . . . 11 dom (𝐹𝐴) ⊆ 𝐴
3433sseli 3817 . . . . . . . . . 10 (𝑦 ∈ dom (𝐹𝐴) → 𝑦𝐴)
3531, 34syl6 35 . . . . . . . . 9 ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦𝐴))
3635expcomd 408 . . . . . . . 8 ((Smo 𝐹 ∧ Ord 𝐴) → (𝑥 ∈ dom (𝐹𝐴) → (𝑦𝑥𝑦𝐴)))
3736imp31 410 . . . . . . 7 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → 𝑦𝐴)
3837fvresd 6466 . . . . . 6 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
3933sseli 3817 . . . . . . . 8 (𝑥 ∈ dom (𝐹𝐴) → 𝑥𝐴)
4039fvresd 6466 . . . . . . 7 (𝑥 ∈ dom (𝐹𝐴) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4140ad2antlr 717 . . . . . 6 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4238, 41eleq12d 2853 . . . . 5 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → (((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ (𝐹𝑦) ∈ (𝐹𝑥)))
4342ralbidva 3167 . . . 4 (((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) → (∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ ∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
4443ralbidva 3167 . . 3 ((Smo 𝐹 ∧ Ord 𝐴) → (∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
4529, 44mpbird 249 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥))
46 dfsmo2 7727 . 2 (Smo (𝐹𝐴) ↔ ((𝐹𝐴):dom (𝐹𝐴)⟶On ∧ Ord dom (𝐹𝐴) ∧ ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥)))
4714, 22, 45, 46syl3anbrc 1400 1 ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090   ∩ cin 3791   ⊆ wss 3792  dom cdm 5355  ran crn 5356   ↾ cres 5357   “ cima 5358  Ord word 5975  Oncon0 5976  Fun wfun 6129   Fn wfn 6130  ⟶wf 6131  ‘cfv 6135  Smo wsmo 7725 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-smo 7726 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator