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Theorem issmo 6546
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
StepHypRef Expression
1 issmo.1 . . 3  |-  A : B
--> On
2 issmo.4 . . . 4  |-  dom  A  =  B
32feq2i 5526 . . 3  |-  ( A : dom  A --> On  <->  A : B
--> On )
41, 3mpbir 201 . 2  |-  A : dom  A --> On
5 issmo.2 . . 3  |-  Ord  B
6 ordeq 4529 . . . 4  |-  ( dom 
A  =  B  -> 
( Ord  dom  A  <->  Ord  B ) )
72, 6ax-mp 8 . . 3  |-  ( Ord 
dom  A  <->  Ord  B )
85, 7mpbir 201 . 2  |-  Ord  dom  A
92eleq2i 2451 . . . 4  |-  ( x  e.  dom  A  <->  x  e.  B )
102eleq2i 2451 . . . 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 ) ) )
129, 10, 11syl2anb 466 . . 3  |-  ( ( x  e.  dom  A  /\  y  e.  dom  A )  ->  ( x  e.  y  ->  ( A `
 x )  e.  ( A `  y
) ) )
1312rgen2a 2715 . 2  |-  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) )
14 df-smo 6544 . 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
) ) ) )
154, 8, 13, 14mpbir3an 1136 1  |-  Smo  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   Ord word 4521   Oncon0 4522   dom cdm 4818   -->wf 5390   ` cfv 5394   Smo wsmo 6543
This theorem is referenced by:  iordsmo  6555  smobeth  8394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-in 3270  df-ss 3277  df-uni 3958  df-tr 4244  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-fn 5397  df-f 5398  df-smo 6544
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