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Theorem issmo 7020
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 5724 . . 3  |-  ( A : dom  A --> On  <->  A : B
--> On )
41, 3mpbir 209 . 2  |-  A : dom  A --> On
5 issmo.2 . . 3  |-  Ord  B
6 ordeq 4885 . . . 4  |-  ( dom 
A  =  B  -> 
( Ord  dom  A  <->  Ord  B ) )
72, 6ax-mp 5 . . 3  |-  ( Ord 
dom  A  <->  Ord  B )
85, 7mpbir 209 . 2  |-  Ord  dom  A
92eleq2i 2545 . . . 4  |-  ( x  e.  dom  A  <->  x  e.  B )
102eleq2i 2545 . . . 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 479 . . 3  |-  ( ( x  e.  dom  A  /\  y  e.  dom  A )  ->  ( x  e.  y  ->  ( A `
 x )  e.  ( A `  y
) ) )
1312rgen2a 2891 . 2  |-  A. x  e.  dom  A A. y  e.  dom  A ( x  e.  y  ->  ( A `  x )  e.  ( A `  y
) )
14 df-smo 7018 . 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 1178 1  |-  Smo  A
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   Ord word 4877   Oncon0 4878   dom cdm 4999   -->wf 5584   ` cfv 5588   Smo wsmo 7017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-in 3483  df-ss 3490  df-uni 4246  df-tr 4541  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-fn 5591  df-f 5592  df-smo 7018
This theorem is referenced by:  iordsmo  7029  smobeth  8962
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