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Theorem isf32lem1 7979
Description: Lemma for isfin3-2 7993. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)
Hypotheses
Ref Expression
isf32lem.a  |-  ( ph  ->  F : om --> ~P G
)
isf32lem.b  |-  ( ph  ->  A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x ) )
isf32lem.c  |-  ( ph  ->  -.  |^| ran  F  e. 
ran  F )
Assertion
Ref Expression
isf32lem1  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  C_  A  /\  ph ) )  -> 
( F `  A
)  C_  ( F `  B ) )
Distinct variable groups:    x, B    ph, x    x, A    x, F
Allowed substitution hint:    G( x)

Proof of Theorem isf32lem1
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . 5  |-  ( a  =  B  ->  ( F `  a )  =  ( F `  B ) )
21sseq1d 3205 . . . 4  |-  ( a  =  B  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  B )  C_  ( F `  B )
) )
32imbi2d 307 . . 3  |-  ( a  =  B  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  B ) 
C_  ( F `  B ) ) ) )
4 fveq2 5525 . . . . 5  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
54sseq1d 3205 . . . 4  |-  ( a  =  b  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  b )  C_  ( F `  B )
) )
65imbi2d 307 . . 3  |-  ( a  =  b  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  b ) 
C_  ( F `  B ) ) ) )
7 fveq2 5525 . . . . 5  |-  ( a  =  suc  b  -> 
( F `  a
)  =  ( F `
 suc  b )
)
87sseq1d 3205 . . . 4  |-  ( a  =  suc  b  -> 
( ( F `  a )  C_  ( F `  B )  <->  ( F `  suc  b
)  C_  ( F `  B ) ) )
98imbi2d 307 . . 3  |-  ( a  =  suc  b  -> 
( ( ph  ->  ( F `  a ) 
C_  ( F `  B ) )  <->  ( ph  ->  ( F `  suc  b )  C_  ( F `  B )
) ) )
10 fveq2 5525 . . . . 5  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
1110sseq1d 3205 . . . 4  |-  ( a  =  A  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  A )  C_  ( F `  B )
) )
1211imbi2d 307 . . 3  |-  ( a  =  A  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  A ) 
C_  ( F `  B ) ) ) )
13 ssid 3197 . . . 4  |-  ( F `
 B )  C_  ( F `  B )
1413a1ii 24 . . 3  |-  ( B  e.  om  ->  ( ph  ->  ( F `  B )  C_  ( F `  B )
) )
15 isf32lem.b . . . . . . 7  |-  ( ph  ->  A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x ) )
16 suceq 4457 . . . . . . . . . 10  |-  ( x  =  b  ->  suc  x  =  suc  b )
1716fveq2d 5529 . . . . . . . . 9  |-  ( x  =  b  ->  ( F `  suc  x )  =  ( F `  suc  b ) )
18 fveq2 5525 . . . . . . . . 9  |-  ( x  =  b  ->  ( F `  x )  =  ( F `  b ) )
1917, 18sseq12d 3207 . . . . . . . 8  |-  ( x  =  b  ->  (
( F `  suc  x )  C_  ( F `  x )  <->  ( F `  suc  b
)  C_  ( F `  b ) ) )
2019rspcv 2880 . . . . . . 7  |-  ( b  e.  om  ->  ( A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x )  ->  ( F `  suc  b ) 
C_  ( F `  b ) ) )
2115, 20syl5 28 . . . . . 6  |-  ( b  e.  om  ->  ( ph  ->  ( F `  suc  b )  C_  ( F `  b )
) )
2221ad2antrr 706 . . . . 5  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( ph  ->  ( F `  suc  b
)  C_  ( F `  b ) ) )
23 sstr2 3186 . . . . 5  |-  ( ( F `  suc  b
)  C_  ( F `  b )  ->  (
( F `  b
)  C_  ( F `  B )  ->  ( F `  suc  b ) 
C_  ( F `  B ) ) )
2422, 23syl6 29 . . . 4  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( ph  ->  ( ( F `  b
)  C_  ( F `  B )  ->  ( F `  suc  b ) 
C_  ( F `  B ) ) ) )
2524a2d 23 . . 3  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( ( ph  ->  ( F `  b
)  C_  ( F `  B ) )  -> 
( ph  ->  ( F `
 suc  b )  C_  ( F `  B
) ) ) )
263, 6, 9, 12, 14, 25findsg 4683 . 2  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( ph  ->  ( F `  A ) 
C_  ( F `  B ) ) )
2726impr 602 1  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  C_  A  /\  ph ) )  -> 
( F `  A
)  C_  ( F `  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ~Pcpw 3625   |^|cint 3862   suc csuc 4394   omcom 4656   ran crn 4690   -->wf 5251   ` cfv 5255
This theorem is referenced by:  isf32lem2  7980  isf32lem3  7981
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-iota 5219  df-fv 5263
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