Theorem smores3 6058

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 4734	. . . . . 6 |- dom ( A |` B ) = ( B i^i dom A )
2		incom 3192	. . . . . 6 |- ( B i^i dom A ) = ( dom A i^i B )
3	1, 2	eqtri 2108	. . . . 5 |- dom ( A |` B ) = ( dom A i^i B )
4	3	eleq2i 2154	. . . 4 |- ( C e. dom ( A |` B ) <-> C e. ( dom A i^i B ) )
5		smores 6057	. . . 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 963	. 2 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> Smo ( ( A |` B ) |` C ) )
8		inss2 3221	. . . . . 6 |- ( dom A i^i B ) C_ B
9	8	sseli 3021	. . . . 5 |- ( C e. ( dom A i^i B ) -> C e. B )
10		ordelss 4206	. . . . . 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 961	. . 3 |- ( ( Smo ( A |` B ) /\ C e. ( dom A i^i B ) /\ Ord B ) -> C C_ B )
14		resabs1 4742	. . 3 |- ( C C_ B -> ( ( A |` B ) |` C ) = ( A |` C ) )
15		smoeq 6055	. . 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 924   = wceq 1289   e. wcel 1438   i^i cin 2998   C_ wss 2999   Ord word 4189   dom cdm 4438   |` cres 4440   Smo wsmo 6050

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-tr 3937  df-iord 4193  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-smo 6051

This theorem is referenced by: (None)