Theorem smores3 6564

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 5084	. . . . . 6 |- dom ( A |` B ) = ( B i^i dom A )
2		incom 3421	. . . . . 6 |- ( B i^i dom A ) = ( dom A i^i B )
3	1, 2	eqtri 2259	. . . . 5 |- dom ( A |` B ) = ( dom A i^i B )
4	3	eleq2i 2305	. . . 4 |- ( C e. dom ( A |` B ) <-> C e. ( dom A i^i B ) )
5		smores 6563	. . . 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 1048	. 2 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo ( ( A |` B ) |` C ) )
8		inss2 3452	. . . . . 6 |- ( dom A i^i B ) C_ B
9	8	sseli 3244	. . . . 5 |- ( C e. ( dom A i^i B ) -> C e. B )
10		ordelss 4524	. . . . . 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 1046	. . 3 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> C C_ B )
14		resabs1 5092	. . 3 |- ( C C_ B -> ( ( A |` B ) |` C ) = ( A |` C ) )
15		smoeq 6561	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   i^i cin 3219   C_ wss 3220   Ord word 4507   dom cdm 4774   |` cres 4776   Smo wsmo 6556

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-tr 4230  df-iord 4511  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-smo 6557

This theorem is used by: (None)