Theorem smores3 5931

Description: A strictly monotone function restricted to an ordinal remains strictly monotone. (Contributed by Andrew Salmon, 19-Nov-2011.)

Assertion

Ref	Expression
smores3	|- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo ( A |` C ) )

Proof of Theorem smores3

Step	Hyp	Ref	Expression
1		dmres 4650	. . . . . 6 |- dom ( A |` B ) = ( B i^i dom A )
2		incom 3158	. . . . . 6 |- ( B i^i dom A ) = ( dom A i^i B )
3	1, 2	eqtri 2101	. . . . 5 |- dom ( A |` B ) = ( dom A i^i B )
4	3	eleq2i 2145	. . . 4 |- ( C e. dom ( A |` B ) <-> C e. ( dom A i^i B ) )
5		smores 5930	. . . 4 |- ( ( Smo ( A |` B ) /\ C e. dom ( A |` B ) ) -> Smo ( ( A |` B ) |` C ) )
6	4, 5	sylan2br 282	. . 3 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) ) -> Smo ( ( A |` B ) |` C ) )
7	6	3adant3 958	. 2 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo ( ( A |` B ) |` C ) )
8		inss2 3187	. . . . . 6 |- ( dom A i^i B ) C_ B
9	8	sseli 2995	. . . . 5 |- ( C e. ( dom A i^i B ) -> C e. B )
10		ordelss 4134	. . . . . 6 |- ( ( Ord B /\ C e. B ) -> C C_ B )
11	10	ancoms 264	. . . . 5 |- ( ( C e. B /\ Ord B ) -> C C_ B )
12	9, 11	sylan 277	. . . 4 |- ( ( C e. ( dom A i^i B ) /\ Ord B ) -> C C_ B )
13	12	3adant1 956	. . 3 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> C C_ B )
14		resabs1 4658	. . 3 |- ( C C_ B -> ( ( A |` B ) |` C ) = ( A |` C ) )
15		smoeq 5928	. . 3 |- ( ( ( A |` B ) |` C ) = ( A |` C ) -> ( Smo ( ( A |` B ) |` C ) <-> Smo ( A |` C ) ) )
16	13, 14, 15	3syl 17	. 2 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> ( Smo ( ( A |` B ) |` C ) <-> Smo ( A |` C ) ) )
17	7, 16	mpbid 145	1 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo ( A |` C ) )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   i^i cin 2972   C_ wss 2973   Ord word 4117   dom cdm 4363   |` cres 4365   Smo wsmo 5923

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-tr 3876  df-iord 4121  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-smo 5924

This theorem is referenced by: (None)