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Theorem isf32lem1 8772
Description: Lemma for isfin3-2 8786. 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 5872 . . . . 5  |-  ( a  =  B  ->  ( F `  a )  =  ( F `  B ) )
21sseq1d 3488 . . . 4  |-  ( a  =  B  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  B )  C_  ( F `  B )
) )
32imbi2d 317 . . 3  |-  ( a  =  B  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  B ) 
C_  ( F `  B ) ) ) )
4 fveq2 5872 . . . . 5  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
54sseq1d 3488 . . . 4  |-  ( a  =  b  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  b )  C_  ( F `  B )
) )
65imbi2d 317 . . 3  |-  ( a  =  b  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  b ) 
C_  ( F `  B ) ) ) )
7 fveq2 5872 . . . . 5  |-  ( a  =  suc  b  -> 
( F `  a
)  =  ( F `
 suc  b )
)
87sseq1d 3488 . . . 4  |-  ( a  =  suc  b  -> 
( ( F `  a )  C_  ( F `  B )  <->  ( F `  suc  b
)  C_  ( F `  B ) ) )
98imbi2d 317 . . 3  |-  ( a  =  suc  b  -> 
( ( ph  ->  ( F `  a ) 
C_  ( F `  B ) )  <->  ( ph  ->  ( F `  suc  b )  C_  ( F `  B )
) ) )
10 fveq2 5872 . . . . 5  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
1110sseq1d 3488 . . . 4  |-  ( a  =  A  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  A )  C_  ( F `  B )
) )
1211imbi2d 317 . . 3  |-  ( a  =  A  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  A ) 
C_  ( F `  B ) ) ) )
13 ssid 3480 . . . 4  |-  ( F `
 B )  C_  ( F `  B )
14132a1i 12 . . 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 5498 . . . . . . . . . 10  |-  ( x  =  b  ->  suc  x  =  suc  b )
1716fveq2d 5876 . . . . . . . . 9  |-  ( x  =  b  ->  ( F `  suc  x )  =  ( F `  suc  b ) )
18 fveq2 5872 . . . . . . . . 9  |-  ( x  =  b  ->  ( F `  x )  =  ( F `  b ) )
1917, 18sseq12d 3490 . . . . . . . 8  |-  ( x  =  b  ->  (
( F `  suc  x )  C_  ( F `  x )  <->  ( F `  suc  b
)  C_  ( F `  b ) ) )
2019rspcv 3175 . . . . . . 7  |-  ( b  e.  om  ->  ( A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x )  ->  ( F `  suc  b ) 
C_  ( F `  b ) ) )
2115, 20syl5 33 . . . . . 6  |-  ( b  e.  om  ->  ( ph  ->  ( F `  suc  b )  C_  ( F `  b )
) )
2221ad2antrr 730 . . . . 5  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( ph  ->  ( F `  suc  b
)  C_  ( F `  b ) ) )
23 sstr2 3468 . . . . 5  |-  ( ( F `  suc  b
)  C_  ( F `  b )  ->  (
( F `  b
)  C_  ( F `  B )  ->  ( F `  suc  b ) 
C_  ( F `  B ) ) )
2422, 23syl6 34 . . . 4  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( ph  ->  ( ( F `  b
)  C_  ( F `  B )  ->  ( F `  suc  b ) 
C_  ( F `  B ) ) ) )
2524a2d 29 . . 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 6725 . 2  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( ph  ->  ( F `  A ) 
C_  ( F `  B ) ) )
2726impr 623 1  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  ( B  C_  A  /\  ph ) )  -> 
( F `  A
)  C_  ( F `  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   A.wral 2773    C_ wss 3433   ~Pcpw 3976   |^|cint 4249   ran crn 4846   suc csuc 5435   -->wf 5588   ` cfv 5592   omcom 6697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fv 5600  df-om 6698
This theorem is referenced by:  isf32lem2  8773  isf32lem3  8774
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