Theorem smores3 6439

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 5026	. . . . . 6 |- dom ( A |` B ) = ( B i^i dom A )
2		incom 3396	. . . . . 6 |- ( B i^i dom A ) = ( dom A i^i B )
3	1, 2	eqtri 2250	. . . . 5 |- dom ( A |` B ) = ( dom A i^i B )
4	3	eleq2i 2296	. . . 4 |- ( C e. dom ( A |` B ) <-> C e. ( dom A i^i B ) )
5		smores 6438	. . . 4 |- ( ( Smo ( A |` B ) /\ C e. dom ( A |` B ) ) -> Smo ( ( A |` B ) |` C ) )
6	4, 5	sylan2br 288	. . 3 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) ) -> Smo ( ( A |` B ) |` C ) )
7	6	3adant3 1041	. 2 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo ( ( A |` B ) |` C ) )
8		inss2 3425	. . . . . 6 |- ( dom A i^i B ) C_ B
9	8	sseli 3220	. . . . 5 |- ( C e. ( dom A i^i B ) -> C e. B )
10		ordelss 4470	. . . . . 6 |- ( ( Ord B /\ C e. B ) -> C C_ B )
11	10	ancoms 268	. . . . 5 |- ( ( C e. B /\ Ord B ) -> C C_ B )
12	9, 11	sylan 283	. . . 4 |- ( ( C e. ( dom A i^i B ) /\ Ord B ) -> C C_ B )
13	12	3adant1 1039	. . 3 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> C C_ B )
14		resabs1 5034	. . 3 |- ( C C_ B -> ( ( A |` B ) |` C ) = ( A |` C ) )
15		smoeq 6436	. . 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 147	1 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo ( A |` C ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   i^i cin 3196   C_ wss 3197   Ord word 4453   dom cdm 4719   |` cres 4721   Smo wsmo 6431

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-tr 4183  df-iord 4457  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-smo 6432

This theorem is referenced by: (None)