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Theorem smores 7569
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 6042 . . . . . . . 8 (Fun 𝐴 → Fun (𝐴𝐵))
2 funfn 6031 . . . . . . . 8 (Fun 𝐴𝐴 Fn dom 𝐴)
3 funfn 6031 . . . . . . . 8 (Fun (𝐴𝐵) ↔ (𝐴𝐵) Fn dom (𝐴𝐵))
41, 2, 33imtr3i 280 . . . . . . 7 (𝐴 Fn dom 𝐴 → (𝐴𝐵) Fn dom (𝐴𝐵))
5 resss 5532 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
6 rnss 5461 . . . . . . . . 9 ((𝐴𝐵) ⊆ 𝐴 → ran (𝐴𝐵) ⊆ ran 𝐴)
75, 6ax-mp 5 . . . . . . . 8 ran (𝐴𝐵) ⊆ ran 𝐴
8 sstr 3717 . . . . . . . 8 ((ran (𝐴𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ⊆ On) → ran (𝐴𝐵) ⊆ On)
97, 8mpan 708 . . . . . . 7 (ran 𝐴 ⊆ On → ran (𝐴𝐵) ⊆ On)
104, 9anim12i 591 . . . . . 6 ((𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On) → ((𝐴𝐵) Fn dom (𝐴𝐵) ∧ ran (𝐴𝐵) ⊆ On))
11 df-f 6005 . . . . . 6 (𝐴:dom 𝐴⟶On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ On))
12 df-f 6005 . . . . . 6 ((𝐴𝐵):dom (𝐴𝐵)⟶On ↔ ((𝐴𝐵) Fn dom (𝐴𝐵) ∧ ran (𝐴𝐵) ⊆ On))
1310, 11, 123imtr4i 281 . . . . 5 (𝐴:dom 𝐴⟶On → (𝐴𝐵):dom (𝐴𝐵)⟶On)
1413a1i 11 . . . 4 (𝐵 ∈ dom 𝐴 → (𝐴:dom 𝐴⟶On → (𝐴𝐵):dom (𝐴𝐵)⟶On))
15 ordelord 5858 . . . . . . 7 ((Ord dom 𝐴𝐵 ∈ dom 𝐴) → Ord 𝐵)
1615expcom 450 . . . . . 6 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord 𝐵))
17 ordin 5866 . . . . . . 7 ((Ord 𝐵 ∧ Ord dom 𝐴) → Ord (𝐵 ∩ dom 𝐴))
1817ex 449 . . . . . 6 (Ord 𝐵 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
1916, 18syli 39 . . . . 5 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord (𝐵 ∩ dom 𝐴)))
20 dmres 5529 . . . . . 6 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
21 ordeq 5843 . . . . . 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 242 . . . 4 (𝐵 ∈ dom 𝐴 → (Ord dom 𝐴 → Ord dom (𝐴𝐵)))
24 dmss 5430 . . . . . . . . 9 ((𝐴𝐵) ⊆ 𝐴 → dom (𝐴𝐵) ⊆ dom 𝐴)
255, 24ax-mp 5 . . . . . . . 8 dom (𝐴𝐵) ⊆ dom 𝐴
26 ssralv 3772 . . . . . . . 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 3772 . . . . . . . . 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 3054 . . . . . . 7 (∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
3127, 30syl 17 . . . . . 6 (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
32 inss1 3941 . . . . . . . . . . . . 13 (𝐵 ∩ dom 𝐴) ⊆ 𝐵
3320, 32eqsstri 3741 . . . . . . . . . . . 12 dom (𝐴𝐵) ⊆ 𝐵
34 simpl 474 . . . . . . . . . . . 12 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑥 ∈ dom (𝐴𝐵))
3533, 34sseldi 3707 . . . . . . . . . . 11 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑥𝐵)
36 fvres 6320 . . . . . . . . . . 11 (𝑥𝐵 → ((𝐴𝐵)‘𝑥) = (𝐴𝑥))
3735, 36syl 17 . . . . . . . . . 10 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑥) = (𝐴𝑥))
38 simpr 479 . . . . . . . . . . . 12 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑦 ∈ dom (𝐴𝐵))
3933, 38sseldi 3707 . . . . . . . . . . 11 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → 𝑦𝐵)
40 fvres 6320 . . . . . . . . . . 11 (𝑦𝐵 → ((𝐴𝐵)‘𝑦) = (𝐴𝑦))
4139, 40syl 17 . . . . . . . . . 10 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝐴𝐵)‘𝑦) = (𝐴𝑦))
4237, 41eleq12d 2797 . . . . . . . . 9 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → (((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦) ↔ (𝐴𝑥) ∈ (𝐴𝑦)))
4342imbi2d 329 . . . . . . . 8 ((𝑥 ∈ dom (𝐴𝐵) ∧ 𝑦 ∈ dom (𝐴𝐵)) → ((𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ (𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
4443ralbidva 3087 . . . . . . 7 (𝑥 ∈ dom (𝐴𝐵) → (∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ ∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
4544ralbiia 3081 . . . . . 6 (∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)) ↔ ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)))
4631, 45sylibr 224 . . . . 5 (∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦)) → ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦)))
4746a1i 11 . . . 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 1519 . . 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 7563 . . 3 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟶On ∧ Ord dom 𝐴 ∧ ∀𝑥 ∈ dom 𝐴𝑦 ∈ dom 𝐴(𝑥𝑦 → (𝐴𝑥) ∈ (𝐴𝑦))))
50 df-smo 7563 . . 3 (Smo (𝐴𝐵) ↔ ((𝐴𝐵):dom (𝐴𝐵)⟶On ∧ Ord dom (𝐴𝐵) ∧ ∀𝑥 ∈ dom (𝐴𝐵)∀𝑦 ∈ dom (𝐴𝐵)(𝑥𝑦 → ((𝐴𝐵)‘𝑥) ∈ ((𝐴𝐵)‘𝑦))))
5148, 49, 503imtr4g 285 . 2 (𝐵 ∈ dom 𝐴 → (Smo 𝐴 → Smo (𝐴𝐵)))
5251impcom 445 1 ((Smo 𝐴𝐵 ∈ dom 𝐴) → Smo (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  cin 3679  wss 3680  dom cdm 5218  ran crn 5219  cres 5220  Ord word 5835  Oncon0 5836  Fun wfun 5995   Fn wfn 5996  wf 5997  cfv 6001  Smo wsmo 7562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-tr 4861  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ord 5839  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-smo 7563
This theorem is referenced by:  smores3  7570  alephsing  9211
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