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Theorem isf32lem1 8734
Description: Lemma for isfin3-2 8748. 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 5866 . . . . 5  |-  ( a  =  B  ->  ( F `  a )  =  ( F `  B ) )
21sseq1d 3531 . . . 4  |-  ( a  =  B  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  B )  C_  ( F `  B )
) )
32imbi2d 316 . . 3  |-  ( a  =  B  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  B ) 
C_  ( F `  B ) ) ) )
4 fveq2 5866 . . . . 5  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
54sseq1d 3531 . . . 4  |-  ( a  =  b  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  b )  C_  ( F `  B )
) )
65imbi2d 316 . . 3  |-  ( a  =  b  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  b ) 
C_  ( F `  B ) ) ) )
7 fveq2 5866 . . . . 5  |-  ( a  =  suc  b  -> 
( F `  a
)  =  ( F `
 suc  b )
)
87sseq1d 3531 . . . 4  |-  ( a  =  suc  b  -> 
( ( F `  a )  C_  ( F `  B )  <->  ( F `  suc  b
)  C_  ( F `  B ) ) )
98imbi2d 316 . . 3  |-  ( a  =  suc  b  -> 
( ( ph  ->  ( F `  a ) 
C_  ( F `  B ) )  <->  ( ph  ->  ( F `  suc  b )  C_  ( F `  B )
) ) )
10 fveq2 5866 . . . . 5  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
1110sseq1d 3531 . . . 4  |-  ( a  =  A  ->  (
( F `  a
)  C_  ( F `  B )  <->  ( F `  A )  C_  ( F `  B )
) )
1211imbi2d 316 . . 3  |-  ( a  =  A  ->  (
( ph  ->  ( F `
 a )  C_  ( F `  B ) )  <->  ( ph  ->  ( F `  A ) 
C_  ( F `  B ) ) ) )
13 ssid 3523 . . . 4  |-  ( F `
 B )  C_  ( F `  B )
1413a1ii 27 . . 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 4943 . . . . . . . . . 10  |-  ( x  =  b  ->  suc  x  =  suc  b )
1716fveq2d 5870 . . . . . . . . 9  |-  ( x  =  b  ->  ( F `  suc  x )  =  ( F `  suc  b ) )
18 fveq2 5866 . . . . . . . . 9  |-  ( x  =  b  ->  ( F `  x )  =  ( F `  b ) )
1917, 18sseq12d 3533 . . . . . . . 8  |-  ( x  =  b  ->  (
( F `  suc  x )  C_  ( F `  x )  <->  ( F `  suc  b
)  C_  ( F `  b ) ) )
2019rspcv 3210 . . . . . . 7  |-  ( b  e.  om  ->  ( A. x  e.  om  ( F `  suc  x
)  C_  ( F `  x )  ->  ( F `  suc  b ) 
C_  ( F `  b ) ) )
2115, 20syl5 32 . . . . . 6  |-  ( b  e.  om  ->  ( ph  ->  ( F `  suc  b )  C_  ( F `  b )
) )
2221ad2antrr 725 . . . . 5  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( ph  ->  ( F `  suc  b
)  C_  ( F `  b ) ) )
23 sstr2 3511 . . . . 5  |-  ( ( F `  suc  b
)  C_  ( F `  b )  ->  (
( F `  b
)  C_  ( F `  B )  ->  ( F `  suc  b ) 
C_  ( F `  B ) ) )
2422, 23syl6 33 . . . 4  |-  ( ( ( b  e.  om  /\  B  e.  om )  /\  B  C_  b )  ->  ( ph  ->  ( ( F `  b
)  C_  ( F `  B )  ->  ( F `  suc  b ) 
C_  ( F `  B ) ) ) )
2524a2d 26 . . 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 6712 . 2  |-  ( ( ( A  e.  om  /\  B  e.  om )  /\  B  C_  A )  ->  ( ph  ->  ( F `  A ) 
C_  ( F `  B ) ) )
2726impr 619 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 369    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   ~Pcpw 4010   |^|cint 4282   suc csuc 4880   ran crn 5000   -->wf 5584   ` cfv 5588   omcom 6685
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-iota 5551  df-fv 5596  df-om 6686
This theorem is referenced by:  isf32lem2  8735  isf32lem3  8736
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