Theorem issmo 5926

Description: Conditions for which A is a strictly monotone ordinal function. (Contributed by Andrew Salmon, 15-Nov-2011.)

Hypotheses

Ref	Expression
issmo.1	|- A : B --> On
issmo.2	|- Ord B
issmo.3	|- ( ( x e. B /\ y e. B ) -> ( x e. y -> ( A ` x ) e. ( A ` y ) ) )
issmo.4	|- dom A = B

Assertion

Ref	Expression
issmo	|- Smo A

Distinct variable group:   x, y, A

Allowed substitution hints:   B( x, y)

Proof of Theorem issmo

Step	Hyp	Ref	Expression
1		issmo.1	. . 3 |- A : B --> On
2		issmo.4	. . . 4 |- dom A = B
3	2	feq2i 5060	. . 3 |- ( A : dom A --> On <-> A : B --> On )
4	1, 3	mpbir 144	. 2 |- A : dom A --> On
5		issmo.2	. . 3 |- Ord B
6		ordeq 4127	. . . 4 |- ( dom A = B -> ( Ord dom A <-> Ord B ) )
7	2, 6	ax-mp 7	. . 3 |- ( Ord dom A <-> Ord B )
8	5, 7	mpbir 144	. 2 |- Ord dom A
9	2	eleq2i 2145	. . . 4 |- ( x e. dom A <-> x e. B )
10	2	eleq2i 2145	. . . 4 |- ( y e. dom A <-> y e. B )
11		issmo.3	. . . 4 |- ( ( x e. B /\ y e. B ) -> ( x e. y -> ( A ` x ) e. ( A ` y ) ) )
12	9, 10, 11	syl2anb 285	. . 3 |- ( ( x e. dom A /\ y e. dom A ) -> ( x e. y -> ( A ` x ) e. ( A ` y ) ) )
13	12	rgen2a 2417	. 2 |- A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) )
14		df-smo 5924	. 2 |- ( Smo A <-> ( A : dom A --> On /\ Ord dom A /\ A. x e. dom A A. y e. dom A ( x e. y -> ( A ` x ) e. ( A ` y ) ) ) )
15	4, 8, 13, 14	mpbir3an 1120	1 |- Smo A

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   Ord word 4117   Oncon0 4118   dom cdm 4363   -->wf 4918   ` cfv 4922   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-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

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-in 2979  df-ss 2986  df-uni 3602  df-tr 3876  df-iord 4121  df-fn 4925  df-f 4926  df-smo 5924

This theorem is referenced by:  iordsmo  5935