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Theorem isf32lem1 7995
Description: Lemma for isfin3-2 8009. 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 5541 . . . . 5  |-  ( a  =  B  ->  ( F `  a )  =  ( F `  B ) )
21sseq1d 3218 . . . 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 5541 . . . . 5  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
54sseq1d 3218 . . . 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 5541 . . . . 5  |-  ( a  =  suc  b  -> 
( F `  a
)  =  ( F `
 suc  b )
)
87sseq1d 3218 . . . 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 5541 . . . . 5  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
1110sseq1d 3218 . . . 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 3210 . . . 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 4473 . . . . . . . . . 10  |-  ( x  =  b  ->  suc  x  =  suc  b )
1716fveq2d 5545 . . . . . . . . 9  |-  ( x  =  b  ->  ( F `  suc  x )  =  ( F `  suc  b ) )
18 fveq2 5541 . . . . . . . . 9  |-  ( x  =  b  ->  ( F `  x )  =  ( F `  b ) )
1917, 18sseq12d 3220 . . . . . . . 8  |-  ( x  =  b  ->  (
( F `  suc  x )  C_  ( F `  x )  <->  ( F `  suc  b
)  C_  ( F `  b ) ) )
2019rspcv 2893 . . . . . . 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 3199 . . . . 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 4699 . 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 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ~Pcpw 3638   |^|cint 3878   suc csuc 4410   omcom 4672   ran crn 4706   -->wf 5267   ` cfv 5271
This theorem is referenced by:  isf32lem2  7996  isf32lem3  7997
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-iota 5235  df-fv 5279
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