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Theorem smores 8299
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 6544 . . . . . . . 8 (Fun 𝐴 β†’ Fun (𝐴 β†Ύ 𝐡))
2 funfn 6532 . . . . . . . 8 (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3 funfn 6532 . . . . . . . 8 (Fun (𝐴 β†Ύ 𝐡) ↔ (𝐴 β†Ύ 𝐡) Fn dom (𝐴 β†Ύ 𝐡))
41, 2, 33imtr3i 291 . . . . . . 7 (𝐴 Fn dom 𝐴 β†’ (𝐴 β†Ύ 𝐡) Fn dom (𝐴 β†Ύ 𝐡))
5 resss 5963 . . . . . . . . 9 (𝐴 β†Ύ 𝐡) βŠ† 𝐴
65rnssi 5896 . . . . . . . 8 ran (𝐴 β†Ύ 𝐡) βŠ† ran 𝐴
7 sstr 3953 . . . . . . . 8 ((ran (𝐴 β†Ύ 𝐡) βŠ† ran 𝐴 ∧ ran 𝐴 βŠ† On) β†’ ran (𝐴 β†Ύ 𝐡) βŠ† On)
86, 7mpan 689 . . . . . . 7 (ran 𝐴 βŠ† On β†’ ran (𝐴 β†Ύ 𝐡) βŠ† On)
94, 8anim12i 614 . . . . . 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 6501 . . . . . 6 (𝐴:dom 𝐴⟢On ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 βŠ† On))
11 df-f 6501 . . . . . 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 292 . . . . 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 6340 . . . . . . 7 ((Ord dom 𝐴 ∧ 𝐡 ∈ dom 𝐴) β†’ Ord 𝐡)
1514expcom 415 . . . . . 6 (𝐡 ∈ dom 𝐴 β†’ (Ord dom 𝐴 β†’ Ord 𝐡))
16 ordin 6348 . . . . . . 7 ((Ord 𝐡 ∧ Ord dom 𝐴) β†’ Ord (𝐡 ∩ dom 𝐴))
1716ex 414 . . . . . 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 5960 . . . . . 6 dom (𝐴 β†Ύ 𝐡) = (𝐡 ∩ dom 𝐴)
20 ordeq 6325 . . . . . 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 252 . . . 4 (𝐡 ∈ dom 𝐴 β†’ (Ord dom 𝐴 β†’ Ord dom (𝐴 β†Ύ 𝐡)))
23 dmss 5859 . . . . . . . . 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 4011 . . . . . . . 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 4011 . . . . . . . . 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 4189 . . . . . . . . . . . . 13 (𝐡 ∩ dom 𝐴) βŠ† 𝐡
3219, 31eqsstri 3979 . . . . . . . . . . . 12 dom (𝐴 β†Ύ 𝐡) βŠ† 𝐡
33 simpl 484 . . . . . . . . . . . 12 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ π‘₯ ∈ dom (𝐴 β†Ύ 𝐡))
3432, 33sselid 3943 . . . . . . . . . . 11 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ π‘₯ ∈ 𝐡)
3534fvresd 6863 . . . . . . . . . 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 486 . . . . . . . . . . . 12 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡))
3732, 36sselid 3943 . . . . . . . . . . 11 ((π‘₯ ∈ dom (𝐴 β†Ύ 𝐡) ∧ 𝑦 ∈ dom (𝐴 β†Ύ 𝐡)) β†’ 𝑦 ∈ 𝐡)
3837fvresd 6863 . . . . . . . . . 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 2828 . . . . . . . . 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 341 . . . . . . . 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 3169 . . . . . . 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 1444 . . 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 8293 . . 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 8293 . . 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 296 . 2 (𝐡 ∈ dom 𝐴 β†’ (Smo 𝐴 β†’ Smo (𝐴 β†Ύ 𝐡)))
4948impcom 409 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   ∩ cin 3910   βŠ† wss 3911  dom cdm 5634  ran crn 5635   β†Ύ cres 5636  Ord word 6317  Oncon0 6318  Fun wfun 6491   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  Smo wsmo 8292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-tr 5224  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ord 6321  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-smo 8293
This theorem is referenced by:  smores3  8300  alephsing  10217
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