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Theorem smores 6095
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 5089 . . . . . . . 8 (Fun 𝐴 → Fun (𝐴𝐵))
2 funfn 5079 . . . . . . . 8 (Fun 𝐴𝐴 Fn dom 𝐴)
3 funfn 5079 . . . . . . . 8 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
41, 2, 33imtr3i 199 . . . . . . 7 (𝐴 Fn dom 𝐴 → (𝐴𝐵) Fn dom (𝐴𝐵))
5 resss 4769 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
6 rnss 4697 . . . . . . . . 9 ((𝐴𝐵) ⊆ 𝐴 → ran (𝐴𝐵) ⊆ ran 𝐴)
75, 6ax-mp 7 . . . . . . . 8 ran (𝐴𝐵) ⊆ ran 𝐴
8 sstr 3047 . . . . . . . 8 ((ran (𝐴𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ⊆ On) → ran (𝐴𝐵) ⊆ On)
97, 8mpan 416 . . . . . . 7 (ran 𝐴 ⊆ On → ran (𝐴𝐵) ⊆ On)
104, 9anim12i 332 . . . . . 6 ((𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On) → ((𝐴𝐵) Fn dom (𝐴𝐵) ∧ ran (𝐴𝐵) ⊆ On))
11 df-f 5053 . . . . . 6 (𝐴:dom 𝐴⟶On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On))
12 df-f 5053 . . . . . 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 4232 . . . . . . 7 ((Ord dom 𝐴𝐵 ∈ dom 𝐴) → Ord 𝐵)
1615expcom 115 . . . . . 6 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord 𝐵))
17 ordin 4236 . . . . . . 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 4766 . . . . . 6 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
21 ordeq 4223 . . . . . 6 (dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴) → (Ord dom (𝐴𝐵) ↔ Ord (𝐵 ∩ dom 𝐴)))
2220, 21ax-mp 7 . . . . 5 (Ord dom (𝐴𝐵) ↔ Ord (𝐵 ∩ dom 𝐴))
2319, 22syl6ibr 161 . . . 4 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord dom (𝐴𝐵)))
24 dmss 4666 . . . . . . . . 9 ((𝐴𝐵) ⊆ 𝐴 → dom (𝐴𝐵) ⊆ dom 𝐴)
255, 24ax-mp 7 . . . . . . . 8 dom (𝐴𝐵) ⊆ dom 𝐴
26 ssralv 3100 . . . . . . . 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 7 . . . . . . 7 (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
28 ssralv 3100 . . . . . . . . 9 (dom (𝐴𝐵) ⊆ dom 𝐴 → (∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
2925, 28ax-mp 7 . . . . . . . 8 (∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
3029ralimi 2449 . . . . . . 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 3235 . . . . . . . . . . . . 13 (𝐵 ∩ dom 𝐴) ⊆ 𝐵
3320, 32eqsstri 3071 . . . . . . . . . . . 12 dom (𝐴𝐵) ⊆ 𝐵
34 simpl 108 . . . . . . . . . . . 12 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑥 ∈ dom (𝐴𝐵))
3533, 34sseldi 3037 . . . . . . . . . . 11 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑥𝐵)
36 fvres 5364 . . . . . . . . . . 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 3037 . . . . . . . . . . 11 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑦𝐵)
40 fvres 5364 . . . . . . . . . . 11 (𝑦𝐵 → ((𝐴𝐵)‘𝑦) = (𝐴𝑦))
4139, 40syl 14 . . . . . . . . . 10 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑦) = (𝐴𝑦))
4237, 41eleq12d 2165 . . . . . . . . 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 2387 . . . . . . 7 (𝑥 ∈ dom (𝐴𝐵) → (∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ ∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
4544ralbiia 2403 . . . . . 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 1262 . . 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 6089 . . 3 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
50 df-smo 6089 . . 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 927   = wceq 1296  wcel 1445  wral 2370  cin 3012  wss 3013  Ord word 4213  Oncon0 4214  dom cdm 4467  ran crn 4468  cres 4469  Fun wfun 5043   Fn wfn 5044  wf 5045  cfv 5049  Smo wsmo 6088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-tr 3959  df-iord 4217  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057  df-smo 6089
This theorem is referenced by:  smores3  6096
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