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Theorem issmo 6602
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 5578 . . 3  |-  ( A : dom  A --> On  <->  A : B
--> On )
41, 3mpbir 201 . 2  |-  A : dom  A --> On
5 issmo.2 . . 3  |-  Ord  B
6 ordeq 4580 . . . 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 2499 . . . 4  |-  ( x  e.  dom  A  <->  x  e.  B )
102eleq2i 2499 . . . 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 2764 . 2  |-  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) )
14 df-smo 6600 . 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 1652    e. wcel 1725   A.wral 2697   Ord word 4572   Oncon0 4573   dom cdm 4870   -->wf 5442   ` cfv 5446   Smo wsmo 6599
This theorem is referenced by:  iordsmo  6611  smobeth  8453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-in 3319  df-ss 3326  df-uni 4008  df-tr 4295  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-fn 5449  df-f 5450  df-smo 6600
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