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Theorem smores2 6200
 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 6193 . . . . . . 7 (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
21simp1bi 997 . . . . . 6 (Smo 𝐹𝐹:dom 𝐹⟶On)
3 ffun 5284 . . . . . 6 (𝐹:dom 𝐹⟶On → Fun 𝐹)
42, 3syl 14 . . . . 5 (Smo 𝐹 → Fun 𝐹)
5 funres 5173 . . . . . 6 (Fun 𝐹 → Fun (𝐹𝐴))
6 funfn 5162 . . . . . 6 (Fun (𝐹𝐴) ↔ (𝐹𝐴) Fn dom (𝐹𝐴))
75, 6sylib 121 . . . . 5 (Fun 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
84, 7syl 14 . . . 4 (Smo 𝐹 → (𝐹𝐴) Fn dom (𝐹𝐴))
9 df-ima 4561 . . . . . 6 (𝐹𝐴) = ran (𝐹𝐴)
10 imassrn 4901 . . . . . 6 (𝐹𝐴) ⊆ ran 𝐹
119, 10eqsstrri 3136 . . . . 5 ran (𝐹𝐴) ⊆ ran 𝐹
12 frn 5290 . . . . . 6 (𝐹:dom 𝐹⟶On → ran 𝐹 ⊆ On)
132, 12syl 14 . . . . 5 (Smo 𝐹 → ran 𝐹 ⊆ On)
1411, 13sstrid 3114 . . . 4 (Smo 𝐹 → ran (𝐹𝐴) ⊆ On)
15 df-f 5136 . . . 4 ((𝐹𝐴):dom (𝐹𝐴)⟶On ↔ ((𝐹𝐴) Fn dom (𝐹𝐴) ∧ ran (𝐹𝐴) ⊆ On))
168, 14, 15sylanbrc 414 . . 3 (Smo 𝐹 → (𝐹𝐴):dom (𝐹𝐴)⟶On)
1716adantr 274 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → (𝐹𝐴):dom (𝐹𝐴)⟶On)
18 smodm 6197 . . 3 (Smo 𝐹 → Ord dom 𝐹)
19 ordin 4316 . . . . 5 ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord (𝐴 ∩ dom 𝐹))
20 dmres 4849 . . . . . 6 dom (𝐹𝐴) = (𝐴 ∩ dom 𝐹)
21 ordeq 4303 . . . . . 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 133 . . . 4 ((Ord 𝐴 ∧ Ord dom 𝐹) → Ord dom (𝐹𝐴))
2423ancoms 266 . . 3 ((Ord dom 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹𝐴))
2518, 24sylan 281 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → Ord dom (𝐹𝐴))
26 resss 4852 . . . . . 6 (𝐹𝐴) ⊆ 𝐹
27 dmss 4747 . . . . . 6 ((𝐹𝐴) ⊆ 𝐹 → dom (𝐹𝐴) ⊆ dom 𝐹)
2826, 27ax-mp 5 . . . . 5 dom (𝐹𝐴) ⊆ dom 𝐹
291simp3bi 999 . . . . 5 (Smo 𝐹 → ∀𝑥 ∈ dom 𝐹𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
30 ssralv 3167 . . . . 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 274 . . 3 ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥))
33 ordtr1 4319 . . . . . . . . . . 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 3302 . . . . . . . . . . . 12 (𝐴 ∩ dom 𝐹) ⊆ 𝐴
3620, 35eqsstri 3135 . . . . . . . . . . 11 dom (𝐹𝐴) ⊆ 𝐴
3736sseli 3099 . . . . . . . . . 10 (𝑦 ∈ dom (𝐹𝐴) → 𝑦𝐴)
3834, 37syl6 33 . . . . . . . . 9 ((Smo 𝐹 ∧ Ord 𝐴) → ((𝑦𝑥𝑥 ∈ dom (𝐹𝐴)) → 𝑦𝐴))
3938expcomd 1418 . . . . . . . 8 ((Smo 𝐹 ∧ Ord 𝐴) → (𝑥 ∈ dom (𝐹𝐴) → (𝑦𝑥𝑦𝐴)))
4039imp31 254 . . . . . . 7 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → 𝑦𝐴)
41 fvres 5454 . . . . . . 7 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
4240, 41syl 14 . . . . . 6 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
4336sseli 3099 . . . . . . . 8 (𝑥 ∈ dom (𝐹𝐴) → 𝑥𝐴)
44 fvres 5454 . . . . . . . 8 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4543, 44syl 14 . . . . . . 7 (𝑥 ∈ dom (𝐹𝐴) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4645ad2antlr 481 . . . . . 6 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
4742, 46eleq12d 2211 . . . . 5 ((((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) ∧ 𝑦𝑥) → (((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ (𝐹𝑦) ∈ (𝐹𝑥)))
4847ralbidva 2435 . . . 4 (((Smo 𝐹 ∧ Ord 𝐴) ∧ 𝑥 ∈ dom (𝐹𝐴)) → (∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ ∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
4948ralbidva 2435 . . 3 ((Smo 𝐹 ∧ Ord 𝐴) → (∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥) ↔ ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 (𝐹𝑦) ∈ (𝐹𝑥)))
5032, 49mpbird 166 . 2 ((Smo 𝐹 ∧ Ord 𝐴) → ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥))
51 dfsmo2 6193 . 2 (Smo (𝐹𝐴) ↔ ((𝐹𝐴):dom (𝐹𝐴)⟶On ∧ Ord dom (𝐹𝐴) ∧ ∀𝑥 ∈ dom (𝐹𝐴)∀𝑦𝑥 ((𝐹𝐴)‘𝑦) ∈ ((𝐹𝐴)‘𝑥)))
5217, 25, 50, 51syl3anbrc 1166 1 ((Smo 𝐹 ∧ Ord 𝐴) → Smo (𝐹𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417   ∩ cin 3076   ⊆ wss 3077  Ord word 4293  Oncon0 4294  dom cdm 4548  ran crn 4549   ↾ cres 4550   “ cima 4551  Fun wfun 5126   Fn wfn 5127  ⟶wf 5128  ‘cfv 5132  Smo wsmo 6191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-tr 4036  df-iord 4297  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-fv 5140  df-smo 6192 This theorem is referenced by: (None)
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