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Theorem smores 6189
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 5164 . . . . . . . 8 (Fun 𝐴 → Fun (𝐴𝐵))
2 funfn 5153 . . . . . . . 8 (Fun 𝐴𝐴 Fn dom 𝐴)
3 funfn 5153 . . . . . . . 8 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
41, 2, 33imtr3i 199 . . . . . . 7 (𝐴 Fn dom 𝐴 → (𝐴𝐵) Fn dom (𝐴𝐵))
5 resss 4843 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
6 rnss 4769 . . . . . . . . 9 ((𝐴𝐵) ⊆ 𝐴 → ran (𝐴𝐵) ⊆ ran 𝐴)
75, 6ax-mp 5 . . . . . . . 8 ran (𝐴𝐵) ⊆ ran 𝐴
8 sstr 3105 . . . . . . . 8 ((ran (𝐴𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ⊆ On) → ran (𝐴𝐵) ⊆ On)
97, 8mpan 420 . . . . . . 7 (ran 𝐴 ⊆ On → ran (𝐴𝐵) ⊆ On)
104, 9anim12i 336 . . . . . 6 ((𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On) → ((𝐴𝐵) Fn dom (𝐴𝐵) ∧ ran (𝐴𝐵) ⊆ On))
11 df-f 5127 . . . . . 6 (𝐴:dom 𝐴⟶On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On))
12 df-f 5127 . . . . . 6 ((𝐴𝐵):dom (𝐴𝐵)⟶On ↔ ((𝐴𝐵) Fn dom (𝐴𝐵) ∧ ran (𝐴𝐵) ⊆ On))
1310, 11, 123imtr4i 200 . . . . 5 (𝐴:dom 𝐴⟶On → (𝐴𝐵):dom (𝐴𝐵)⟶On)
1413a1i 9 . . . 4 (𝐵 ∈ dom 𝐴 → (𝐴:dom 𝐴⟶On → (𝐴𝐵):dom (𝐴𝐵)⟶On))
15 ordelord 4303 . . . . . . 7 ((Ord dom 𝐴𝐵 ∈ dom 𝐴) → Ord 𝐵)
1615expcom 115 . . . . . 6 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord 𝐵))
17 ordin 4307 . . . . . . 7 ((Ord 𝐵 ∧ Ord dom 𝐴) → Ord (𝐵 ∩ dom 𝐴))
1817ex 114 . . . . . 6 (Ord 𝐵 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
1916, 18syli 37 . . . . 5 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
20 dmres 4840 . . . . . 6 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
21 ordeq 4294 . . . . . 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 161 . . . 4 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord dom (𝐴𝐵)))
24 dmss 4738 . . . . . . . . 9 ((𝐴𝐵) ⊆ 𝐴 → dom (𝐴𝐵) ⊆ dom 𝐴)
255, 24ax-mp 5 . . . . . . . 8 dom (𝐴𝐵) ⊆ dom 𝐴
26 ssralv 3161 . . . . . . . 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 3161 . . . . . . . . 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 2495 . . . . . . 7 (∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
3127, 30syl 14 . . . . . 6 (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
32 inss1 3296 . . . . . . . . . . . . 13 (𝐵 ∩ dom 𝐴) ⊆ 𝐵
3320, 32eqsstri 3129 . . . . . . . . . . . 12 dom (𝐴𝐵) ⊆ 𝐵
34 simpl 108 . . . . . . . . . . . 12 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑥 ∈ dom (𝐴𝐵))
3533, 34sseldi 3095 . . . . . . . . . . 11 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑥𝐵)
36 fvres 5445 . . . . . . . . . . 11 (𝑥𝐵 → ((𝐴𝐵)‘𝑥) = (𝐴𝑥))
3735, 36syl 14 . . . . . . . . . 10 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑥) = (𝐴𝑥))
38 simpr 109 . . . . . . . . . . . 12 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑦 ∈ dom (𝐴𝐵))
3933, 38sseldi 3095 . . . . . . . . . . 11 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑦𝐵)
40 fvres 5445 . . . . . . . . . . 11 (𝑦𝐵 → ((𝐴𝐵)‘𝑦) = (𝐴𝑦))
4139, 40syl 14 . . . . . . . . . 10 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑦) = (𝐴𝑦))
4237, 41eleq12d 2210 . . . . . . . . 9 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → (((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦) ↔ (𝐴𝑥) ∈ (𝐴𝑦)))
4342imbi2d 229 . . . . . . . 8 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
4443ralbidva 2433 . . . . . . 7 (𝑥 ∈ dom (𝐴𝐵) → (∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ ∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
4544ralbiia 2449 . . . . . 6 (∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
4631, 45sylibr 133 . . . . 5 (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)))
4746a1i 9 . . . 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 1297 . . 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 6183 . . 3 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
50 df-smo 6183 . . 3 (Smo (𝐴𝐵) ↔ ((𝐴𝐵):dom (𝐴𝐵)⟶On ∧ Ord dom (𝐴𝐵) ∧ ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦))))
5148, 49, 503imtr4g 204 . 2 (𝐵 ∈ dom 𝐴 → (Smo 𝐴 → Smo (𝐴𝐵)))
5251impcom 124 1 ((Smo 𝐴𝐵 ∈ dom 𝐴) → Smo (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 962   = wceq 1331  wcel 1480  wral 2416  cin 3070  wss 3071  Ord word 4284  Oncon0 4285  dom cdm 4539  ran crn 4540  cres 4541  Fun wfun 5117   Fn wfn 5118  wf 5119  cfv 5123  Smo wsmo 6182
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-tr 4027  df-iord 4288  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-smo 6183
This theorem is referenced by:  smores3  6190
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