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Theorem smores 6295
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 𝐴 ∧ 𝐡 ∈ dom 𝐴) β†’ Smo (𝐴 β†Ύ 𝐡))
Proof of Theorem smores
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funres 5259 . . . . . . . 8 (Fun 𝐴 β†’ Fun (𝐴 β†Ύ 𝐡))
2 funfn 5248 . . . . . . . 8 (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3 funfn 5248 . . . . . . . 8 (Fun (𝐴 β†Ύ 𝐡) ↔ (𝐴 β†Ύ 𝐡) Fn dom (𝐴 β†Ύ 𝐡))
41, 2, 33imtr3i 200 . . . . . . 7 (𝐴 Fn dom 𝐴 β†’ (𝐴 β†Ύ 𝐡) Fn dom (𝐴 β†Ύ 𝐡))
5 resss 4933 . . . . . . . . 9 (𝐴 β†Ύ 𝐡) βŠ† 𝐴
6 rnss 4859 . . . . . . . . 9 ((𝐴 β†Ύ 𝐡) βŠ† 𝐴 β†’ ran (𝐴 β†Ύ 𝐡) βŠ† ran 𝐴)
75, 6ax-mp 5 . . . . . . . 8 ran (𝐴 β†Ύ 𝐡) βŠ† ran 𝐴
8 sstr 3165 . . . . . . . 8 ((ran (𝐴 β†Ύ 𝐡) βŠ† ran 𝐴 ∧ ran 𝐴 βŠ† On) β†’ ran (𝐴 β†Ύ 𝐡) βŠ† On)
97, 8mpan 424 . . . . . . 7 (ran 𝐴 βŠ† On β†’ ran (𝐴 β†Ύ 𝐡) βŠ† On)
104, 9anim12i 338 . . . . . 6 ((𝐴 Fn dom 𝐴 ∧ ran 𝐴 βŠ† On) β†’ ((𝐴 β†Ύ 𝐡) Fn dom (𝐴 β†Ύ 𝐡) ∧ ran (𝐴 β†Ύ 𝐡) βŠ† On))
11 df-f 5222 . . . . . 6 (𝐴:dom 𝐴⟢On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 βŠ† On))
12 df-f 5222 . . . . . 6 ((𝐴 β†Ύ 𝐡):dom (𝐴 β†Ύ 𝐡)⟢On ↔ ((𝐴 β†Ύ 𝐡) Fn dom (𝐴 β†Ύ 𝐡) ∧ ran (𝐴 β†Ύ 𝐡) βŠ† On))
1310, 11, 123imtr4i 201 . . . . 5 (𝐴:dom 𝐴⟢On β†’ (𝐴 β†Ύ 𝐡):dom (𝐴 β†Ύ 𝐡)⟢On)
1413a1i 9 . . . 4 (𝐡 ∈ dom 𝐴 β†’ (𝐴:dom 𝐴⟢On β†’ (𝐴 β†Ύ 𝐡):dom (𝐴 β†Ύ 𝐡)⟢On))
15 ordelord 4383 . . . . . . 7 ((Ord dom 𝐴 ∧ 𝐡 ∈ dom 𝐴) β†’ Ord 𝐡)
1615expcom 116 . . . . . 6 (𝐡 ∈ dom 𝐴 β†’ (Ord dom 𝐴 β†’ Ord 𝐡))
17 ordin 4387 . . . . . . 7 ((Ord 𝐡 ∧ Ord dom 𝐴) β†’ Ord (𝐡 ∩ dom 𝐴))
1817ex 115 . . . . . 6 (Ord 𝐡 β†’ (Ord dom 𝐴 β†’ Ord (𝐡 ∩ dom 𝐴)))
1916, 18syli 37 . . . . 5 (𝐡 ∈ dom 𝐴 β†’ (Ord dom 𝐴 β†’ Ord (𝐡 ∩ dom 𝐴)))
20 dmres 4930 . . . . . 6 dom (𝐴 β†Ύ 𝐡) = (𝐡 ∩ dom 𝐴)
21 ordeq 4374 . . . . . 6 (dom (𝐴 β†Ύ 𝐡) = (𝐡 ∩ dom 𝐴) β†’ (Ord dom (𝐴 β†Ύ 𝐡) ↔ Ord (𝐡 ∩ dom 𝐴)))
2220, 21ax-mp 5 . . . . 5 (Ord dom (𝐴 β†Ύ 𝐡) ↔ Ord (𝐡 ∩ dom 𝐴))
2319, 22imbitrrdi 162 . . . 4 (𝐡 ∈ dom 𝐴 β†’ (Ord dom 𝐴 β†’ Ord dom (𝐴 β†Ύ 𝐡)))
24 dmss 4828 . . . . . . . . 9 ((𝐴 β†Ύ 𝐡) βŠ† 𝐴 β†’ dom (𝐴 β†Ύ 𝐡) βŠ† dom 𝐴)
255, 24ax-mp 5 . . . . . . . 8 dom (𝐴 β†Ύ 𝐡) βŠ† dom 𝐴
26 ssralv 3221 . . . . . . . 8 (dom (𝐴 β†Ύ 𝐡) βŠ† dom 𝐴 β†’ (βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) β†’ βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦))))
2725, 26ax-mp 5 . . . . . . 7 (βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) β†’ βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)))
28 ssralv 3221 . . . . . . . . 9 (dom (𝐴 β†Ύ 𝐡) βŠ† dom 𝐴 β†’ (βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) β†’ βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦))))
2925, 28ax-mp 5 . . . . . . . 8 (βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) β†’ βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)))
3029ralimi 2540 . . . . . . 7 (βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) β†’ βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)))
3127, 30syl 14 . . . . . 6 (βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) β†’ βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)))
32 inss1 3357 . . . . . . . . . . . . 13 (𝐡 ∩ dom 𝐴) βŠ† 𝐡
3320, 32eqsstri 3189 . . . . . . . . . . . 12 dom (𝐴 β†Ύ 𝐡) βŠ† 𝐡
34 simpl 109 . . . . . . . . . . . 12 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ π‘₯ ∈ dom (𝐴 β†Ύ 𝐡))
3533, 34sselid 3155 . . . . . . . . . . 11 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ π‘₯ ∈ 𝐡)
36 fvres 5541 . . . . . . . . . . 11 (π‘₯ ∈ 𝐡 β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘₯) = (π΄β€˜π‘₯))
3735, 36syl 14 . . . . . . . . . 10 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘₯) = (π΄β€˜π‘₯))
38 simpr 110 . . . . . . . . . . . 12 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡))
3933, 38sselid 3155 . . . . . . . . . . 11 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ 𝑦 ∈ 𝐡)
40 fvres 5541 . . . . . . . . . . 11 (𝑦 ∈ 𝐡 β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘¦) = (π΄β€˜π‘¦))
4139, 40syl 14 . . . . . . . . . 10 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘¦) = (π΄β€˜π‘¦))
4237, 41eleq12d 2248 . . . . . . . . 9 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ (((𝐴 β†Ύ 𝐡)β€˜π‘₯) ∈ ((𝐴 β†Ύ 𝐡)β€˜π‘¦) ↔ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)))
4342imbi2d 230 . . . . . . . 8 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ ((π‘₯ ∈ 𝑦 β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘₯) ∈ ((𝐴 β†Ύ 𝐡)β€˜π‘¦)) ↔ (π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦))))
4443ralbidva 2473 . . . . . . 7 (π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) β†’ (βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘₯) ∈ ((𝐴 β†Ύ 𝐡)β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦))))
4544ralbiia 2491 . . . . . 6 (βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘₯) ∈ ((𝐴 β†Ύ 𝐡)β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)))
4631, 45sylibr 134 . . . . 5 (βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) β†’ βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘₯) ∈ ((𝐴 β†Ύ 𝐡)β€˜π‘¦)))
4746a1i 9 . . . 4 (𝐡 ∈ dom 𝐴 β†’ (βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦)) β†’ βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘₯) ∈ ((𝐴 β†Ύ 𝐡)β€˜π‘¦))))
4814, 23, 473anim123d 1319 . . 3 (𝐡 ∈ dom 𝐴 β†’ ((𝐴:dom 𝐴⟢On ∧ Ord dom 𝐴 ∧ βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦))) β†’ ((𝐴 β†Ύ 𝐡):dom (𝐴 β†Ύ 𝐡)⟢On ∧ Ord dom (𝐴 β†Ύ 𝐡) ∧ βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘₯) ∈ ((𝐴 β†Ύ 𝐡)β€˜π‘¦)))))
49 df-smo 6289 . . 3 (Smo 𝐴 ↔ (𝐴:dom 𝐴⟢On ∧ Ord dom 𝐴 ∧ βˆ€π‘₯ ∈ dom π΄βˆ€π‘¦ ∈ dom 𝐴(π‘₯ ∈ 𝑦 β†’ (π΄β€˜π‘₯) ∈ (π΄β€˜π‘¦))))
50 df-smo 6289 . . 3 (Smo (𝐴 β†Ύ 𝐡) ↔ ((𝐴 β†Ύ 𝐡):dom (𝐴 β†Ύ 𝐡)⟢On ∧ Ord dom (𝐴 β†Ύ 𝐡) ∧ βˆ€π‘₯ ∈ dom (𝐴 β†Ύ 𝐡)βˆ€π‘¦ ∈ dom (𝐴 β†Ύ 𝐡)(π‘₯ ∈ 𝑦 β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘₯) ∈ ((𝐴 β†Ύ 𝐡)β€˜π‘¦))))
5148, 49, 503imtr4g 205 . 2 (𝐡 ∈ dom 𝐴 β†’ (Smo 𝐴 β†’ Smo (𝐴 β†Ύ 𝐡)))
5251impcom 125 1 ((Smo 𝐴 ∧ 𝐡 ∈ dom 𝐴) β†’ Smo (𝐴 β†Ύ 𝐡))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455   ∩ cin 3130   βŠ† wss 3131  Ord word 4364  Oncon0 4365  dom cdm 4628  ran crn 4629   β†Ύ cres 4630  Fun wfun 5212   Fn wfn 5213  βŸΆwf 5214  β€˜cfv 5218  Smo wsmo 6288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-tr 4104  df-iord 4368  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-smo 6289
This theorem is referenced by:  smores3  6296
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