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Theorem smores 5848
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 B 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 4884 . . . . . . . 8 (Fun A → Fun (AB))
2 funfn 4874 . . . . . . . 8 (Fun AA Fn dom A)
3 funfn 4874 . . . . . . . 8 (Fun (AB) ↔ (AB) Fn dom (AB))
41, 2, 33imtr3i 189 . . . . . . 7 (A Fn dom A → (AB) Fn dom (AB))
5 resss 4578 . . . . . . . . 9 (AB) ⊆ A
6 rnss 4507 . . . . . . . . 9 ((AB) ⊆ A → ran (AB) ⊆ ran A)
75, 6ax-mp 7 . . . . . . . 8 ran (AB) ⊆ ran A
8 sstr 2947 . . . . . . . 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 4849 . . . . . 6 (A:dom A⟶On ↔ (A Fn dom A ran A ⊆ On))
12 df-f 4849 . . . . . 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 4084 . . . . . . 7 ((Ord dom A B dom A) → Ord B)
1615expcom 109 . . . . . 6 (B dom A → (Ord dom A → Ord B))
17 ordin 4088 . . . . . . 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 4575 . . . . . 6 dom (AB) = (B ∩ dom A)
21 ordeq 4075 . . . . . 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 4477 . . . . . . . . 9 ((AB) ⊆ A → dom (AB) ⊆ dom A)
255, 24ax-mp 7 . . . . . . . 8 dom (AB) ⊆ dom A
26 ssralv 2998 . . . . . . . 8 (dom (AB) ⊆ dom A → (x dom Ay dom A(x y → (Ax) (Ay)) → x dom (AB)y dom A(x y → (Ax) (Ay))))
2725, 26ax-mp 7 . . . . . . 7 (x dom Ay dom A(x y → (Ax) (Ay)) → x dom (AB)y dom A(x y → (Ax) (Ay)))
28 ssralv 2998 . . . . . . . . 9 (dom (AB) ⊆ dom A → (y dom A(x y → (Ax) (Ay)) → y dom (AB)(x y → (Ax) (Ay))))
2925, 28ax-mp 7 . . . . . . . 8 (y dom A(x y → (Ax) (Ay)) → y dom (AB)(x y → (Ax) (Ay)))
3029ralimi 2378 . . . . . . 7 (x dom (AB)y dom A(x y → (Ax) (Ay)) → x dom (AB)y dom (AB)(x y → (Ax) (Ay)))
3127, 30syl 14 . . . . . 6 (x dom Ay dom A(x y → (Ax) (Ay)) → x dom (AB)y dom (AB)(x y → (Ax) (Ay)))
32 inss1 3151 . . . . . . . . . . . . 13 (B ∩ dom A) ⊆ B
3320, 32eqsstri 2969 . . . . . . . . . . . 12 dom (AB) ⊆ B
34 simpl 102 . . . . . . . . . . . 12 ((x dom (AB) y dom (AB)) → x dom (AB))
3533, 34sseldi 2937 . . . . . . . . . . 11 ((x dom (AB) y dom (AB)) → x B)
36 fvres 5141 . . . . . . . . . . 11 (x B → ((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 2937 . . . . . . . . . . 11 ((x dom (AB) y dom (AB)) → y B)
40 fvres 5141 . . . . . . . . . . 11 (y B → ((AB)‘y) = (Ay))
4139, 40syl 14 . . . . . . . . . 10 ((x dom (AB) y dom (AB)) → ((AB)‘y) = (Ay))
4237, 41eleq12d 2105 . . . . . . . . 9 ((x dom (AB) y dom (AB)) → (((AB)‘x) ((AB)‘y) ↔ (Ax) (Ay)))
4342imbi2d 219 . . . . . . . 8 ((x dom (AB) y dom (AB)) → ((x y → ((AB)‘x) ((AB)‘y)) ↔ (x y → (Ax) (Ay))))
4443ralbidva 2316 . . . . . . 7 (x dom (AB) → (y dom (AB)(x y → ((AB)‘x) ((AB)‘y)) ↔ y dom (AB)(x y → (Ax) (Ay))))
4544ralbiia 2332 . . . . . 6 (x dom (AB)y dom (AB)(x y → ((AB)‘x) ((AB)‘y)) ↔ x dom (AB)y dom (AB)(x y → (Ax) (Ay)))
4631, 45sylibr 137 . . . . 5 (x dom Ay dom A(x y → (Ax) (Ay)) → x dom (AB)y dom (AB)(x y → ((AB)‘x) ((AB)‘y)))
4746a1i 9 . . . 4 (B dom A → (x dom Ay dom A(x y → (Ax) (Ay)) → x dom (AB)y dom (AB)(x y → ((AB)‘x) ((AB)‘y))))
4814, 23, 473anim123d 1213 . . 3 (B dom A → ((A:dom A⟶On Ord dom A x dom Ay dom A(x y → (Ax) (Ay))) → ((AB):dom (AB)⟶On Ord dom (AB) x dom (AB)y dom (AB)(x y → ((AB)‘x) ((AB)‘y)))))
49 df-smo 5842 . . 3 (Smo A ↔ (A:dom A⟶On Ord dom A x dom Ay dom A(x y → (Ax) (Ay))))
50 df-smo 5842 . . 3 (Smo (AB) ↔ ((AB):dom (AB)⟶On Ord dom (AB) x dom (AB)y dom (AB)(x y → ((AB)‘x) ((AB)‘y))))
5148, 49, 503imtr4g 194 . 2 (B dom A → (Smo A → Smo (AB)))
5251impcom 116 1 ((Smo A B dom A) → Smo (AB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   = wceq 1242   wcel 1390  wral 2300  cin 2910  wss 2911  Ord word 4065  Oncon0 4066  dom cdm 4288  ran crn 4289  cres 4290  Fun wfun 4839   Fn wfn 4840  wf 4841  cfv 4845  Smo wsmo 5841
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-iord 4069  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-smo 5842
This theorem is referenced by:  smores3  5849
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