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Theorem smores2 7338
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 7331 . . . . . . 7 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
21simp1bi 1069 . . . . . 6 (Smo 𝐹𝐹:dom 𝐹⟶On)
3 ffun 5961 . . . . . 6 (𝐹:dom 𝐹⟶On → Fun 𝐹)
42, 3syl 17 . . . . 5 (Smo 𝐹 → Fun 𝐹)
5 funres 5843 . . . . . 6 (Fun 𝐹 → Fun (𝐹𝐴))
6 funfn 5833 . . . . . 6 (Fun (𝐹𝐴) ↔ (𝐹𝐴) Fn dom (𝐹𝐴))
75, 6sylib 207 . . . . 5 (Fun 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
84, 7syl 17 . . . 4 (Smo 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
9 df-ima 5051 . . . . . 6 (𝐹𝐴) = ran (𝐹𝐴)
10 imassrn 5396 . . . . . 6 (𝐹𝐴) ⊆ ran 𝐹
119, 10eqsstr3i 3599 . . . . 5 ran (𝐹𝐴) ⊆ ran 𝐹
12 frn 5966 . . . . . 6 (𝐹:dom 𝐹⟶On → ran 𝐹 ⊆ On)
132, 12syl 17 . . . . 5 (Smo 𝐹 → ran 𝐹 ⊆ On)
1411, 13syl5ss 3579 . . . 4 (Smo 𝐹 → ran (𝐹𝐴) ⊆ On)
15 df-f 5808 . . . 4 ((𝐹𝐴):dom (𝐹𝐴)⟶On ↔ ((𝐹𝐴) Fn dom (𝐹𝐴) ∧ ran (𝐹𝐴) ⊆ On))
168, 14, 15sylanbrc 695 . . 3 (Smo 𝐹 → (𝐹𝐴):dom (𝐹𝐴)⟶On)
1716adantr 480 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → (𝐹𝐴):dom (𝐹𝐴)⟶On)
18 smodm 7335 . . 3 (Smo 𝐹 → Ord dom 𝐹)
19 ordin 5670 . . . . 5 ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord (𝐴 ∩ dom 𝐹))
20 dmres 5339 . . . . . 6 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
21 ordeq 5647 . . . . . 6 (dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹) → (Ord dom (𝐹𝐴) ↔ Ord (𝐴 ∩ dom 𝐹)))
2220, 21ax-mp 5 . . . . 5 (Ord dom (𝐹𝐴) ↔ Ord (𝐴 ∩ dom 𝐹))
2319, 22sylibr 223 . . . 4 ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord dom (𝐹𝐴))
2423ancoms 468 . . 3 ((Ord dom 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹𝐴))
2518, 24sylan 487 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹𝐴))
26 resss 5342 . . . . . 6 (𝐹𝐴) ⊆ 𝐹
27 dmss 5245 . . . . . 6 ((𝐹𝐴) ⊆ 𝐹 → dom (𝐹𝐴) ⊆ dom 𝐹)
2826, 27ax-mp 5 . . . . 5 dom (𝐹𝐴) ⊆ dom 𝐹
291simp3bi 1071 . . . . 5 (Smo 𝐹 → ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
30 ssralv 3629 . . . . 5 (dom (𝐹𝐴) ⊆ dom 𝐹 → (∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
3128, 29, 30mpsyl 66 . . . 4 (Smo 𝐹 → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
3231adantr 480 . . 3 ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
33 ordtr1 5684 . . . . . . . . . . 11 (Ord dom (𝐹𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦 ∈ dom (𝐹𝐴)))
3425, 33syl 17 . . . . . . . . . 10 ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦 ∈ dom (𝐹𝐴)))
35 inss1 3795 . . . . . . . . . . . 12 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
3620, 35eqsstri 3598 . . . . . . . . . . 11 dom (𝐹𝐴) ⊆ 𝐴
3736sseli 3564 . . . . . . . . . 10 (𝑦 ∈ dom (𝐹𝐴) → 𝑦𝐴)
3834, 37syl6 34 . . . . . . . . 9 ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦𝐴))
3938expcomd 453 . . . . . . . 8 ((Smo 𝐹 ∧ Ord 𝐴) → (𝑥 ∈ dom (𝐹𝐴) → (𝑦𝑥𝑦𝐴)))
4039imp31 447 . . . . . . 7 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → 𝑦𝐴)
41 fvres 6117 . . . . . . 7 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
4240, 41syl 17 . . . . . 6 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
4336sseli 3564 . . . . . . . 8 (𝑥 ∈ dom (𝐹𝐴) → 𝑥𝐴)
44 fvres 6117 . . . . . . . 8 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4543, 44syl 17 . . . . . . 7 (𝑥 ∈ dom (𝐹𝐴) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4645ad2antlr 759 . . . . . 6 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4742, 46eleq12d 2682 . . . . 5 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → (((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ (𝐹𝑦) ∈ (𝐹𝑥)))
4847ralbidva 2968 . . . 4 (((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) → (∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ ∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
4948ralbidva 2968 . . 3 ((Smo 𝐹 ∧ Ord 𝐴) → (∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
5032, 49mpbird 246 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥))
51 dfsmo2 7331 . 2 (Smo (𝐹𝐴) ↔ ((𝐹𝐴):dom (𝐹𝐴)⟶On ∧ Ord dom (𝐹𝐴) ∧ ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥)))
5217, 25, 50, 51syl3anbrc 1239 1 ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  cin 3539  wss 3540  dom cdm 5038  ran crn 5039  cres 5040  cima 5041  Ord word 5639  Oncon0 5640  Fun wfun 5798   Fn wfn 5799  wf 5800  cfv 5804  Smo wsmo 7329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-smo 7330
This theorem is referenced by: (None)
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