Theorem smores3 6272

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 4912	. . . . . 6 |- dom ( A |` B ) = ( B i^i dom A )
2		incom 3319	. . . . . 6 |- ( B i^i dom A ) = ( dom A i^i B )
3	1, 2	eqtri 2191	. . . . 5 |- dom ( A |` B ) = ( dom A i^i B )
4	3	eleq2i 2237	. . . 4 |- ( C e. dom ( A |` B ) <-> C e. ( dom A i^i B ) )
5		smores 6271	. . . 4 |- ( ( Smo ( A |` B ) /\ C e. dom ( A |` B ) ) -> Smo ( ( A |` B ) |` C ) )
6	4, 5	sylan2br 286	. . 3 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) ) -> Smo ( ( A |` B ) |` C ) )
7	6	3adant3 1012	. 2 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo ( ( A |` B ) |` C ) )
8		inss2 3348	. . . . . 6 |- ( dom A i^i B ) C_ B
9	8	sseli 3143	. . . . 5 |- ( C e. ( dom A i^i B ) -> C e. B )
10		ordelss 4364	. . . . . 6 |- ( ( Ord B /\ C e. B ) -> C C_ B )
11	10	ancoms 266	. . . . 5 |- ( ( C e. B /\ Ord B ) -> C C_ B )
12	9, 11	sylan 281	. . . 4 |- ( ( C e. ( dom A i^i B ) /\ Ord B ) -> C C_ B )
13	12	3adant1 1010	. . 3 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> C C_ B )
14		resabs1 4920	. . 3 |- ( C C_ B -> ( ( A |` B ) |` C ) = ( A |` C ) )
15		smoeq 6269	. . 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 146	1 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo ( A |` C ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   i^i cin 3120   C_ wss 3121   Ord word 4347   dom cdm 4611   |` cres 4613   Smo wsmo 6264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-tr 4088  df-iord 4351  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-smo 6265

This theorem is referenced by: (None)