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Theorem smores2 6294
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 6287 . . . . . . 7 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
21simp1bi 1012 . . . . . 6 (Smo 𝐹𝐹:dom 𝐹⟶On)
3 ffun 5368 . . . . . 6 (𝐹:dom 𝐹⟶On → Fun 𝐹)
42, 3syl 14 . . . . 5 (Smo 𝐹 → Fun 𝐹)
5 funres 5257 . . . . . 6 (Fun 𝐹 → Fun (𝐹𝐴))
6 funfn 5246 . . . . . 6 (Fun (𝐹𝐴) ↔ (𝐹𝐴) Fn dom (𝐹𝐴))
75, 6sylib 122 . . . . 5 (Fun 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
84, 7syl 14 . . . 4 (Smo 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
9 df-ima 4639 . . . . . 6 (𝐹𝐴) = ran (𝐹𝐴)
10 imassrn 4981 . . . . . 6 (𝐹𝐴) ⊆ ran 𝐹
119, 10eqsstrri 3188 . . . . 5 ran (𝐹𝐴) ⊆ ran 𝐹
12 frn 5374 . . . . . 6 (𝐹:dom 𝐹⟶On → ran 𝐹 ⊆ On)
132, 12syl 14 . . . . 5 (Smo 𝐹 → ran 𝐹 ⊆ On)
1411, 13sstrid 3166 . . . 4 (Smo 𝐹 → ran (𝐹𝐴) ⊆ On)
15 df-f 5220 . . . 4 ((𝐹𝐴):dom (𝐹𝐴)⟶On ↔ ((𝐹𝐴) Fn dom (𝐹𝐴) ∧ ran (𝐹𝐴) ⊆ On))
168, 14, 15sylanbrc 417 . . 3 (Smo 𝐹 → (𝐹𝐴):dom (𝐹𝐴)⟶On)
1716adantr 276 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → (𝐹𝐴):dom (𝐹𝐴)⟶On)
18 smodm 6291 . . 3 (Smo 𝐹 → Ord dom 𝐹)
19 ordin 4385 . . . . 5 ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord (𝐴 ∩ dom 𝐹))
20 dmres 4928 . . . . . 6 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
21 ordeq 4372 . . . . . 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 134 . . . 4 ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord dom (𝐹𝐴))
2423ancoms 268 . . 3 ((Ord dom 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹𝐴))
2518, 24sylan 283 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹𝐴))
26 resss 4931 . . . . . 6 (𝐹𝐴) ⊆ 𝐹
27 dmss 4826 . . . . . 6 ((𝐹𝐴) ⊆ 𝐹 → dom (𝐹𝐴) ⊆ dom 𝐹)
2826, 27ax-mp 5 . . . . 5 dom (𝐹𝐴) ⊆ dom 𝐹
291simp3bi 1014 . . . . 5 (Smo 𝐹 → ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
30 ssralv 3219 . . . . 5 (dom (𝐹𝐴) ⊆ dom 𝐹 → (∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
3128, 29, 30mpsyl 65 . . . 4 (Smo 𝐹 → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
3231adantr 276 . . 3 ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
33 ordtr1 4388 . . . . . . . . . . 11 (Ord dom (𝐹𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦 ∈ dom (𝐹𝐴)))
3425, 33syl 14 . . . . . . . . . 10 ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦 ∈ dom (𝐹𝐴)))
35 inss1 3355 . . . . . . . . . . . 12 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
3620, 35eqsstri 3187 . . . . . . . . . . 11 dom (𝐹𝐴) ⊆ 𝐴
3736sseli 3151 . . . . . . . . . 10 (𝑦 ∈ dom (𝐹𝐴) → 𝑦𝐴)
3834, 37syl6 33 . . . . . . . . 9 ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦𝐴))
3938expcomd 1441 . . . . . . . 8 ((Smo 𝐹 ∧ Ord 𝐴) → (𝑥 ∈ dom (𝐹𝐴) → (𝑦𝑥𝑦𝐴)))
4039imp31 256 . . . . . . 7 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → 𝑦𝐴)
41 fvres 5539 . . . . . . 7 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
4240, 41syl 14 . . . . . 6 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
4336sseli 3151 . . . . . . . 8 (𝑥 ∈ dom (𝐹𝐴) → 𝑥𝐴)
44 fvres 5539 . . . . . . . 8 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4543, 44syl 14 . . . . . . 7 (𝑥 ∈ dom (𝐹𝐴) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4645ad2antlr 489 . . . . . 6 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4742, 46eleq12d 2248 . . . . 5 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → (((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ (𝐹𝑦) ∈ (𝐹𝑥)))
4847ralbidva 2473 . . . 4 (((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) → (∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ ∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
4948ralbidva 2473 . . 3 ((Smo 𝐹 ∧ Ord 𝐴) → (∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
5032, 49mpbird 167 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥))
51 dfsmo2 6287 . 2 (Smo (𝐹𝐴) ↔ ((𝐹𝐴):dom (𝐹𝐴)⟶On ∧ Ord dom (𝐹𝐴) ∧ ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥)))
5217, 25, 50, 51syl3anbrc 1181 1 ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  cin 3128  wss 3129  Ord word 4362  Oncon0 4363  dom cdm 4626  ran crn 4627  cres 4628  cima 4629  Fun wfun 5210   Fn wfn 5211  wf 5212  cfv 5216  Smo wsmo 6285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-tr 4102  df-iord 4366  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-smo 6286
This theorem is referenced by: (None)
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