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Theorem isf32lem1 8233
Description: Lemma for isfin3-2 8247. 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 5728 . . . . 5  |-  ( a  =  B  ->  ( F `  a )  =  ( F `  B ) )
21sseq1d 3375 . . . 4  |-  ( a  =  B  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  B )  C_  ( F `  B )
) )
32imbi2d 308 . . 3  |-  ( a  =  B  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  B ) 
C_  ( F `  B ) ) ) )
4 fveq2 5728 . . . . 5  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
54sseq1d 3375 . . . 4  |-  ( a  =  b  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  b )  C_  ( F `  B )
) )
65imbi2d 308 . . 3  |-  ( a  =  b  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  b ) 
C_  ( F `  B ) ) ) )
7 fveq2 5728 . . . . 5  |-  ( a  =  suc  b  -> 
( F `  a
)  =  ( F `
 suc  b )
)
87sseq1d 3375 . . . 4  |-  ( a  =  suc  b  -> 
( ( F `  a )  C_  ( F `  B )  <->  ( F `  suc  b
)  C_  ( F `  B ) ) )
98imbi2d 308 . . 3  |-  ( a  =  suc  b  -> 
( ( ph  ->  ( F `  a ) 
C_  ( F `  B ) )  <->  ( ph  ->  ( F `  suc  b )  C_  ( F `  B )
) ) )
10 fveq2 5728 . . . . 5  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
1110sseq1d 3375 . . . 4  |-  ( a  =  A  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  A )  C_  ( F `  B )
) )
1211imbi2d 308 . . 3  |-  ( a  =  A  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  A ) 
C_  ( F `  B ) ) ) )
13 ssid 3367 . . . 4  |-  ( F `
 B )  C_  ( F `  B )
1413a1ii 25 . . 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 4646 . . . . . . . . . 10  |-  ( x  =  b  ->  suc  x  =  suc  b )
1716fveq2d 5732 . . . . . . . . 9  |-  ( x  =  b  ->  ( F `  suc  x )  =  ( F `  suc  b ) )
18 fveq2 5728 . . . . . . . . 9  |-  ( x  =  b  ->  ( F `  x )  =  ( F `  b ) )
1917, 18sseq12d 3377 . . . . . . . 8  |-  ( x  =  b  ->  (
( F `  suc  x )  C_  ( F `  x )  <->  ( F `  suc  b
)  C_  ( F `  b ) ) )
2019rspcv 3048 . . . . . . 7  |-  ( b  e.  om  ->  ( A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x )  ->  ( F `  suc  b ) 
C_  ( F `  b ) ) )
2115, 20syl5 30 . . . . . 6  |-  ( b  e.  om  ->  ( ph  ->  ( F `  suc  b )  C_  ( F `  b )
) )
2221ad2antrr 707 . . . . 5  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( ph  ->  ( F `  suc  b
)  C_  ( F `  b ) ) )
23 sstr2 3355 . . . . 5  |-  ( ( F `  suc  b
)  C_  ( F `  b )  ->  (
( F `  b
)  C_  ( F `  B )  ->  ( F `  suc  b ) 
C_  ( F `  B ) ) )
2422, 23syl6 31 . . . 4  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( ph  ->  ( ( F `  b
)  C_  ( F `  B )  ->  ( F `  suc  b ) 
C_  ( F `  B ) ) ) )
2524a2d 24 . . 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 4872 . 2  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( ph  ->  ( F `  A ) 
C_  ( F `  B ) ) )
2726impr 603 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 359    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   ~Pcpw 3799   |^|cint 4050   suc csuc 4583   omcom 4845   ran crn 4879   -->wf 5450   ` cfv 5454
This theorem is referenced by:  isf32lem2  8234  isf32lem3  8235
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-iota 5418  df-fv 5462
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