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

Theorem smores 7688
Description: A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 16-Nov-2011.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
smores ((Smo 𝐴𝐵 ∈ dom 𝐴) → Smo (𝐴𝐵))

Proof of Theorem smores
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funres 6146 . . . . . . . 8 (Fun 𝐴 → Fun (𝐴𝐵))
2 funfn 6134 . . . . . . . 8 (Fun 𝐴𝐴 Fn dom 𝐴)
3 funfn 6134 . . . . . . . 8 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
41, 2, 33imtr3i 282 . . . . . . 7 (𝐴 Fn dom 𝐴 → (𝐴𝐵) Fn dom (𝐴𝐵))
5 resss 5632 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
6 rnss 5562 . . . . . . . . 9 ((𝐴𝐵) ⊆ 𝐴 → ran (𝐴𝐵) ⊆ ran 𝐴)
75, 6ax-mp 5 . . . . . . . 8 ran (𝐴𝐵) ⊆ ran 𝐴
8 sstr 3813 . . . . . . . 8 ((ran (𝐴𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ⊆ On) → ran (𝐴𝐵) ⊆ On)
97, 8mpan 673 . . . . . . 7 (ran 𝐴 ⊆ On → ran (𝐴𝐵) ⊆ On)
104, 9anim12i 602 . . . . . 6 ((𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On) → ((𝐴𝐵) Fn dom (𝐴𝐵) ∧ ran (𝐴𝐵) ⊆ On))
11 df-f 6108 . . . . . 6 (𝐴:dom 𝐴⟶On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On))
12 df-f 6108 . . . . . 6 ((𝐴𝐵):dom (𝐴𝐵)⟶On ↔ ((𝐴𝐵) Fn dom (𝐴𝐵) ∧ ran (𝐴𝐵) ⊆ On))
1310, 11, 123imtr4i 283 . . . . 5 (𝐴:dom 𝐴⟶On → (𝐴𝐵):dom (𝐴𝐵)⟶On)
1413a1i 11 . . . 4 (𝐵 ∈ dom 𝐴 → (𝐴:dom 𝐴⟶On → (𝐴𝐵):dom (𝐴𝐵)⟶On))
15 ordelord 5965 . . . . . . 7 ((Ord dom 𝐴𝐵 ∈ dom 𝐴) → Ord 𝐵)
1615expcom 400 . . . . . 6 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord 𝐵))
17 ordin 5973 . . . . . . 7 ((Ord 𝐵 ∧ Ord dom 𝐴) → Ord (𝐵 ∩ dom 𝐴))
1817ex 399 . . . . . 6 (Ord 𝐵 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
1916, 18syli 39 . . . . 5 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
20 dmres 5629 . . . . . 6 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
21 ordeq 5950 . . . . . 6 (dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴) → (Ord dom (𝐴𝐵) ↔ Ord (𝐵 ∩ dom 𝐴)))
2220, 21ax-mp 5 . . . . 5 (Ord dom (𝐴𝐵) ↔ Ord (𝐵 ∩ dom 𝐴))
2319, 22syl6ibr 243 . . . 4 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord dom (𝐴𝐵)))
24 dmss 5531 . . . . . . . . 9 ((𝐴𝐵) ⊆ 𝐴 → dom (𝐴𝐵) ⊆ dom 𝐴)
255, 24ax-mp 5 . . . . . . . 8 dom (𝐴𝐵) ⊆ dom 𝐴
26 ssralv 3870 . . . . . . . 8 (dom (𝐴𝐵) ⊆ dom 𝐴 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
2725, 26ax-mp 5 . . . . . . 7 (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
28 ssralv 3870 . . . . . . . . 9 (dom (𝐴𝐵) ⊆ dom 𝐴 → (∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
2925, 28ax-mp 5 . . . . . . . 8 (∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
3029ralimi 3147 . . . . . . 7 (∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
3127, 30syl 17 . . . . . 6 (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
32 inss1 4036 . . . . . . . . . . . . 13 (𝐵 ∩ dom 𝐴) ⊆ 𝐵
3320, 32eqsstri 3839 . . . . . . . . . . . 12 dom (𝐴𝐵) ⊆ 𝐵
34 simpl 470 . . . . . . . . . . . 12 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑥 ∈ dom (𝐴𝐵))
3533, 34sseldi 3803 . . . . . . . . . . 11 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑥𝐵)
36 fvres 6430 . . . . . . . . . . 11 (𝑥𝐵 → ((𝐴𝐵)‘𝑥) = (𝐴𝑥))
3735, 36syl 17 . . . . . . . . . 10 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑥) = (𝐴𝑥))
38 simpr 473 . . . . . . . . . . . 12 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑦 ∈ dom (𝐴𝐵))
3933, 38sseldi 3803 . . . . . . . . . . 11 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑦𝐵)
40 fvres 6430 . . . . . . . . . . 11 (𝑦𝐵 → ((𝐴𝐵)‘𝑦) = (𝐴𝑦))
4139, 40syl 17 . . . . . . . . . 10 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑦) = (𝐴𝑦))
4237, 41eleq12d 2886 . . . . . . . . 9 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → (((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦) ↔ (𝐴𝑥) ∈ (𝐴𝑦)))
4342imbi2d 331 . . . . . . . 8 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
4443ralbidva 3180 . . . . . . 7 (𝑥 ∈ dom (𝐴𝐵) → (∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ ∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
4544ralbiia 3174 . . . . . 6 (∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
4631, 45sylibr 225 . . . . 5 (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)))
4746a1i 11 . . . 4 (𝐵 ∈ dom 𝐴 → (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦))))
4814, 23, 473anim123d 1560 . . 3 (𝐵 ∈ dom 𝐴 → ((𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))) → ((𝐴𝐵):dom (𝐴𝐵)⟶On ∧ Ord dom (𝐴𝐵) ∧ ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)))))
49 df-smo 7682 . . 3 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
50 df-smo 7682 . . 3 (Smo (𝐴𝐵) ↔ ((𝐴𝐵):dom (𝐴𝐵)⟶On ∧ Ord dom (𝐴𝐵) ∧ ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦))))
5148, 49, 503imtr4g 287 . 2 (𝐵 ∈ dom 𝐴 → (Smo 𝐴 → Smo (𝐴𝐵)))
5251impcom 396 1 ((Smo 𝐴𝐵 ∈ dom 𝐴) → Smo (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2157  wral 3103  cin 3775  wss 3776  dom cdm 5318  ran crn 5319  cres 5320  Ord word 5942  Oncon0 5943  Fun wfun 6098   Fn wfn 6099  wf 6100  cfv 6104  Smo wsmo 7681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-tr 4954  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ord 5946  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-fv 6112  df-smo 7682
This theorem is referenced by:  smores3  7689  alephsing  9386
  Copyright terms: Public domain W3C validator