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Theorem smores 5907
 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 4941 . . . . . . . 8 (Fun 𝐴 → Fun (𝐴𝐵))
2 funfn 4931 . . . . . . . 8 (Fun 𝐴𝐴 Fn dom 𝐴)
3 funfn 4931 . . . . . . . 8 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
41, 2, 33imtr3i 189 . . . . . . 7 (𝐴 Fn dom 𝐴 → (𝐴𝐵) Fn dom (𝐴𝐵))
5 resss 4635 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
6 rnss 4564 . . . . . . . . 9 ((𝐴𝐵) ⊆ 𝐴 → ran (𝐴𝐵) ⊆ ran 𝐴)
75, 6ax-mp 7 . . . . . . . 8 ran (𝐴𝐵) ⊆ ran 𝐴
8 sstr 2953 . . . . . . . 8 ((ran (𝐴𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ⊆ On) → ran (𝐴𝐵) ⊆ On)
97, 8mpan 400 . . . . . . 7 (ran 𝐴 ⊆ On → ran (𝐴𝐵) ⊆ On)
104, 9anim12i 321 . . . . . 6 ((𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On) → ((𝐴𝐵) Fn dom (𝐴𝐵) ∧ ran (𝐴𝐵) ⊆ On))
11 df-f 4906 . . . . . 6 (𝐴:dom 𝐴⟶On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On))
12 df-f 4906 . . . . . 6 ((𝐴𝐵):dom (𝐴𝐵)⟶On ↔ ((𝐴𝐵) Fn dom (𝐴𝐵) ∧ ran (𝐴𝐵) ⊆ On))
1310, 11, 123imtr4i 190 . . . . 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 4118 . . . . . . 7 ((Ord dom 𝐴𝐵 ∈ dom 𝐴) → Ord 𝐵)
1615expcom 109 . . . . . 6 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord 𝐵))
17 ordin 4122 . . . . . . 7 ((Ord 𝐵 ∧ Ord dom 𝐴) → Ord (𝐵 ∩ dom 𝐴))
1817ex 108 . . . . . 6 (Ord 𝐵 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
1916, 18syli 33 . . . . 5 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
20 dmres 4632 . . . . . 6 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
21 ordeq 4109 . . . . . 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 151 . . . 4 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord dom (𝐴𝐵)))
24 dmss 4534 . . . . . . . . 9 ((𝐴𝐵) ⊆ 𝐴 → dom (𝐴𝐵) ⊆ dom 𝐴)
255, 24ax-mp 7 . . . . . . . 8 dom (𝐴𝐵) ⊆ dom 𝐴
26 ssralv 3004 . . . . . . . 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 3004 . . . . . . . . 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 2384 . . . . . . 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 3157 . . . . . . . . . . . . 13 (𝐵 ∩ dom 𝐴) ⊆ 𝐵
3320, 32eqsstri 2975 . . . . . . . . . . . 12 dom (𝐴𝐵) ⊆ 𝐵
34 simpl 102 . . . . . . . . . . . 12 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑥 ∈ dom (𝐴𝐵))
3533, 34sseldi 2943 . . . . . . . . . . 11 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑥𝐵)
36 fvres 5198 . . . . . . . . . . 11 (𝑥𝐵 → ((𝐴𝐵)‘𝑥) = (𝐴𝑥))
3735, 36syl 14 . . . . . . . . . 10 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑥) = (𝐴𝑥))
38 simpr 103 . . . . . . . . . . . 12 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑦 ∈ dom (𝐴𝐵))
3933, 38sseldi 2943 . . . . . . . . . . 11 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑦𝐵)
40 fvres 5198 . . . . . . . . . . 11 (𝑦𝐵 → ((𝐴𝐵)‘𝑦) = (𝐴𝑦))
4139, 40syl 14 . . . . . . . . . 10 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑦) = (𝐴𝑦))
4237, 41eleq12d 2108 . . . . . . . . 9 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → (((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦) ↔ (𝐴𝑥) ∈ (𝐴𝑦)))
4342imbi2d 219 . . . . . . . 8 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
4443ralbidva 2322 . . . . . . 7 (𝑥 ∈ dom (𝐴𝐵) → (∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ ∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
4544ralbiia 2338 . . . . . 6 (∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
4631, 45sylibr 137 . . . . 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 1214 . . 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 5901 . . 3 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
50 df-smo 5901 . . 3 (Smo (𝐴𝐵) ↔ ((𝐴𝐵):dom (𝐴𝐵)⟶On ∧ Ord dom (𝐴𝐵) ∧ ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦))))
5148, 49, 503imtr4g 194 . 2 (𝐵 ∈ dom 𝐴 → (Smo 𝐴 → Smo (𝐴𝐵)))
5251impcom 116 1 ((Smo 𝐴𝐵 ∈ dom 𝐴) → Smo (𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∀wral 2306   ∩ cin 2916   ⊆ wss 2917  Ord word 4099  Oncon0 4100  dom cdm 4345  ran crn 4346   ↾ cres 4347  Fun wfun 4896   Fn wfn 4897  ⟶wf 4898  ‘cfv 4902  Smo wsmo 5900 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-tr 3855  df-iord 4103  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-smo 5901 This theorem is referenced by:  smores3  5908
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