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Theorem smores2 8393
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 8386 . . . . . . 7 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
21simp1bi 1144 . . . . . 6 (Smo 𝐹𝐹:dom 𝐹⟶On)
32ffund 6741 . . . . 5 (Smo 𝐹 → Fun 𝐹)
4 funres 6610 . . . . . 6 (Fun 𝐹 → Fun (𝐹𝐴))
54funfnd 6599 . . . . 5 (Fun 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
63, 5syl 17 . . . 4 (Smo 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
7 df-ima 5702 . . . . . 6 (𝐹𝐴) = ran (𝐹𝐴)
8 imassrn 6091 . . . . . 6 (𝐹𝐴) ⊆ ran 𝐹
97, 8eqsstrri 4031 . . . . 5 ran (𝐹𝐴) ⊆ ran 𝐹
102frnd 6745 . . . . 5 (Smo 𝐹 → ran 𝐹 ⊆ On)
119, 10sstrid 4007 . . . 4 (Smo 𝐹 → ran (𝐹𝐴) ⊆ On)
12 df-f 6567 . . . 4 ((𝐹𝐴):dom (𝐹𝐴)⟶On ↔ ((𝐹𝐴) Fn dom (𝐹𝐴) ∧ ran (𝐹𝐴) ⊆ On))
136, 11, 12sylanbrc 583 . . 3 (Smo 𝐹 → (𝐹𝐴):dom (𝐹𝐴)⟶On)
1413adantr 480 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → (𝐹𝐴):dom (𝐹𝐴)⟶On)
15 smodm 8390 . . 3 (Smo 𝐹 → Ord dom 𝐹)
16 ordin 6416 . . . . 5 ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord (𝐴 ∩ dom 𝐹))
17 dmres 6032 . . . . . 6 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
18 ordeq 6393 . . . . . 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 234 . . . 4 ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord dom (𝐹𝐴))
2120ancoms 458 . . 3 ((Ord dom 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹𝐴))
2215, 21sylan 580 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹𝐴))
23 resss 6022 . . . . . 6 (𝐹𝐴) ⊆ 𝐹
24 dmss 5916 . . . . . 6 ((𝐹𝐴) ⊆ 𝐹 → dom (𝐹𝐴) ⊆ dom 𝐹)
2523, 24ax-mp 5 . . . . 5 dom (𝐹𝐴) ⊆ dom 𝐹
261simp3bi 1146 . . . . 5 (Smo 𝐹 → ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
27 ssralv 4064 . . . . 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 480 . . 3 ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
30 ordtr1 6429 . . . . . . . . . . 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 4245 . . . . . . . . . . . 12 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
3317, 32eqsstri 4030 . . . . . . . . . . 11 dom (𝐹𝐴) ⊆ 𝐴
3433sseli 3991 . . . . . . . . . 10 (𝑦 ∈ dom (𝐹𝐴) → 𝑦𝐴)
3531, 34syl6 35 . . . . . . . . 9 ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦𝐴))
3635expcomd 416 . . . . . . . 8 ((Smo 𝐹 ∧ Ord 𝐴) → (𝑥 ∈ dom (𝐹𝐴) → (𝑦𝑥𝑦𝐴)))
3736imp31 417 . . . . . . 7 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → 𝑦𝐴)
3837fvresd 6927 . . . . . 6 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
3933sseli 3991 . . . . . . . 8 (𝑥 ∈ dom (𝐹𝐴) → 𝑥𝐴)
4039fvresd 6927 . . . . . . 7 (𝑥 ∈ dom (𝐹𝐴) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4140ad2antlr 727 . . . . . 6 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4238, 41eleq12d 2833 . . . . 5 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → (((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ (𝐹𝑦) ∈ (𝐹𝑥)))
4342ralbidva 3174 . . . 4 (((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) → (∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ ∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
4443ralbidva 3174 . . 3 ((Smo 𝐹 ∧ Ord 𝐴) → (∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
4529, 44mpbird 257 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥))
46 dfsmo2 8386 . 2 (Smo (𝐹𝐴) ↔ ((𝐹𝐴):dom (𝐹𝐴)⟶On ∧ Ord dom (𝐹𝐴) ∧ ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥)))
4714, 22, 45, 46syl3anbrc 1342 1 ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cin 3962  wss 3963  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Ord word 6385  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  Smo wsmo 8384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-smo 8385
This theorem is referenced by: (None)
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