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Theorem smores 8348
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 6587 . . . . . . . 8 (Fun 𝐴 β†’ Fun (𝐴 β†Ύ 𝐡))
2 funfn 6575 . . . . . . . 8 (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3 funfn 6575 . . . . . . . 8 (Fun (𝐴 β†Ύ 𝐡) ↔ (𝐴 β†Ύ 𝐡) Fn dom (𝐴 β†Ύ 𝐡))
41, 2, 33imtr3i 290 . . . . . . 7 (𝐴 Fn dom 𝐴 β†’ (𝐴 β†Ύ 𝐡) Fn dom (𝐴 β†Ύ 𝐡))
5 resss 6004 . . . . . . . . 9 (𝐴 β†Ύ 𝐡) βŠ† 𝐴
65rnssi 5937 . . . . . . . 8 ran (𝐴 β†Ύ 𝐡) βŠ† ran 𝐴
7 sstr 3989 . . . . . . . 8 ((ran (𝐴 β†Ύ 𝐡) βŠ† ran 𝐴 ∧ ran 𝐴 βŠ† On) β†’ ran (𝐴 β†Ύ 𝐡) βŠ† On)
86, 7mpan 688 . . . . . . 7 (ran 𝐴 βŠ† On β†’ ran (𝐴 β†Ύ 𝐡) βŠ† On)
94, 8anim12i 613 . . . . . 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))
10 df-f 6544 . . . . . 6 (𝐴:dom 𝐴⟢On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 βŠ† On))
11 df-f 6544 . . . . . 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))
129, 10, 113imtr4i 291 . . . . 5 (𝐴:dom 𝐴⟢On β†’ (𝐴 β†Ύ 𝐡):dom (𝐴 β†Ύ 𝐡)⟢On)
1312a1i 11 . . . 4 (𝐡 ∈ dom 𝐴 β†’ (𝐴:dom 𝐴⟢On β†’ (𝐴 β†Ύ 𝐡):dom (𝐴 β†Ύ 𝐡)⟢On))
14 ordelord 6383 . . . . . . 7 ((Ord dom 𝐴 ∧ 𝐡 ∈ dom 𝐴) β†’ Ord 𝐡)
1514expcom 414 . . . . . 6 (𝐡 ∈ dom 𝐴 β†’ (Ord dom 𝐴 β†’ Ord 𝐡))
16 ordin 6391 . . . . . . 7 ((Ord 𝐡 ∧ Ord dom 𝐴) β†’ Ord (𝐡 ∩ dom 𝐴))
1716ex 413 . . . . . 6 (Ord 𝐡 β†’ (Ord dom 𝐴 β†’ Ord (𝐡 ∩ dom 𝐴)))
1815, 17syli 39 . . . . 5 (𝐡 ∈ dom 𝐴 β†’ (Ord dom 𝐴 β†’ Ord (𝐡 ∩ dom 𝐴)))
19 dmres 6001 . . . . . 6 dom (𝐴 β†Ύ 𝐡) = (𝐡 ∩ dom 𝐴)
20 ordeq 6368 . . . . . 6 (dom (𝐴 β†Ύ 𝐡) = (𝐡 ∩ dom 𝐴) β†’ (Ord dom (𝐴 β†Ύ 𝐡) ↔ Ord (𝐡 ∩ dom 𝐴)))
2119, 20ax-mp 5 . . . . 5 (Ord dom (𝐴 β†Ύ 𝐡) ↔ Ord (𝐡 ∩ dom 𝐴))
2218, 21syl6ibr 251 . . . 4 (𝐡 ∈ dom 𝐴 β†’ (Ord dom 𝐴 β†’ Ord dom (𝐴 β†Ύ 𝐡)))
23 dmss 5900 . . . . . . . . 9 ((𝐴 β†Ύ 𝐡) βŠ† 𝐴 β†’ dom (𝐴 β†Ύ 𝐡) βŠ† dom 𝐴)
245, 23ax-mp 5 . . . . . . . 8 dom (𝐴 β†Ύ 𝐡) βŠ† dom 𝐴
25 ssralv 4049 . . . . . . . 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‘¦))))
2624, 25ax-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‘¦)))
27 ssralv 4049 . . . . . . . . 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‘¦))))
2824, 27ax-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‘¦)))
2928ralimi 3083 . . . . . . 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‘¦)))
3026, 29syl 17 . . . . . 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‘¦)))
31 inss1 4227 . . . . . . . . . . . . 13 (𝐡 ∩ dom 𝐴) βŠ† 𝐡
3219, 31eqsstri 4015 . . . . . . . . . . . 12 dom (𝐴 β†Ύ 𝐡) βŠ† 𝐡
33 simpl 483 . . . . . . . . . . . 12 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ π‘₯ ∈ dom (𝐴 β†Ύ 𝐡))
3432, 33sselid 3979 . . . . . . . . . . 11 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ π‘₯ ∈ 𝐡)
3534fvresd 6908 . . . . . . . . . 10 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘₯) = (π΄β€˜π‘₯))
36 simpr 485 . . . . . . . . . . . 12 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡))
3732, 36sselid 3979 . . . . . . . . . . 11 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ 𝑦 ∈ 𝐡)
3837fvresd 6908 . . . . . . . . . 10 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ ((𝐴 β†Ύ 𝐡)β€˜π‘¦) = (π΄β€˜π‘¦))
3935, 38eleq12d 2827 . . . . . . . . 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‘¦)))
4039imbi2d 340 . . . . . . . 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‘¦))))
4140ralbidva 3175 . . . . . . 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‘¦))))
4241ralbiia 3091 . . . . . 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‘¦)))
4330, 42sylibr 233 . . . . 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‘¦)))
4443a1i 11 . . . 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‘¦))))
4513, 22, 443anim123d 1443 . . 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‘¦)))))
46 df-smo 8342 . . 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‘¦))))
47 df-smo 8342 . . 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‘¦))))
4845, 46, 473imtr4g 295 . 2 (𝐡 ∈ dom 𝐴 β†’ (Smo 𝐴 β†’ Smo (𝐴 β†Ύ 𝐡)))
4948impcom 408 1 ((Smo 𝐴 ∧ 𝐡 ∈ dom 𝐴) β†’ Smo (𝐴 β†Ύ 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   ∩ cin 3946   βŠ† wss 3947  dom cdm 5675  ran crn 5676   β†Ύ cres 5677  Ord word 6360  Oncon0 6361  Fun wfun 6534   Fn wfn 6535  βŸΆwf 6536  β€˜cfv 6540  Smo wsmo 8341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ord 6364  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-smo 8342
This theorem is referenced by:  smores3  8349  alephsing  10267
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