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Theorem incssnn0 26786
Description: Transitivity induction of subsets, lemma for nacsfix 26787. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Assertion
Ref Expression
incssnn0  |-  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0  /\  B  e.  ( ZZ>= `  A )
)  ->  ( F `  A )  C_  ( F `  B )
)
Distinct variable group:    x, F
Allowed substitution hints:    A( x)    B( x)

Proof of Theorem incssnn0
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . . . 6  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
21sseq2d 3206 . . . . 5  |-  ( a  =  A  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  A )
) )
32imbi2d 307 . . . 4  |-  ( a  =  A  ->  (
( ( A. x  e.  NN0  ( F `  x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  a )
)  <->  ( ( A. x  e.  NN0  ( F `
 x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  A )
) ) )
4 fveq2 5525 . . . . . 6  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
54sseq2d 3206 . . . . 5  |-  ( a  =  b  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  b )
) )
65imbi2d 307 . . . 4  |-  ( a  =  b  ->  (
( ( A. x  e.  NN0  ( F `  x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  a )
)  <->  ( ( A. x  e.  NN0  ( F `
 x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  b )
) ) )
7 fveq2 5525 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( F `  a )  =  ( F `  ( b  +  1 ) ) )
87sseq2d 3206 . . . . 5  |-  ( a  =  ( b  +  1 )  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  ( b  +  1 ) ) ) )
98imbi2d 307 . . . 4  |-  ( a  =  ( b  +  1 )  ->  (
( ( A. x  e.  NN0  ( F `  x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  a )
)  <->  ( ( A. x  e.  NN0  ( F `
 x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  ( b  +  1 ) ) ) ) )
10 fveq2 5525 . . . . . 6  |-  ( a  =  B  ->  ( F `  a )  =  ( F `  B ) )
1110sseq2d 3206 . . . . 5  |-  ( a  =  B  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  B )
) )
1211imbi2d 307 . . . 4  |-  ( a  =  B  ->  (
( ( A. x  e.  NN0  ( F `  x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  a )
)  <->  ( ( A. x  e.  NN0  ( F `
 x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  B )
) ) )
13 ssid 3197 . . . . 5  |-  ( F `
 A )  C_  ( F `  A )
1413a1ii 24 . . . 4  |-  ( A  e.  ZZ  ->  (
( A. x  e. 
NN0  ( F `  x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  A )
) )
15 eluznn0 10288 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  b  e.  ( ZZ>= `  A ) )  -> 
b  e.  NN0 )
1615ancoms 439 . . . . . . . . 9  |-  ( ( b  e.  ( ZZ>= `  A )  /\  A  e.  NN0 )  ->  b  e.  NN0 )
17 fveq2 5525 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( F `  x )  =  ( F `  b ) )
18 oveq1 5865 . . . . . . . . . . . 12  |-  ( x  =  b  ->  (
x  +  1 )  =  ( b  +  1 ) )
1918fveq2d 5529 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( F `  ( x  +  1 ) )  =  ( F `  ( b  +  1 ) ) )
2017, 19sseq12d 3207 . . . . . . . . . 10  |-  ( x  =  b  ->  (
( F `  x
)  C_  ( F `  ( x  +  1 ) )  <->  ( F `  b )  C_  ( F `  ( b  +  1 ) ) ) )
2120rspcv 2880 . . . . . . . . 9  |-  ( b  e.  NN0  ->  ( A. x  e.  NN0  ( F `
 x )  C_  ( F `  ( x  +  1 ) )  ->  ( F `  b )  C_  ( F `  ( b  +  1 ) ) ) )
2216, 21syl 15 . . . . . . . 8  |-  ( ( b  e.  ( ZZ>= `  A )  /\  A  e.  NN0 )  ->  ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  ->  ( F `  b )  C_  ( F `  (
b  +  1 ) ) ) )
2322expimpd 586 . . . . . . 7  |-  ( b  e.  ( ZZ>= `  A
)  ->  ( ( A  e.  NN0  /\  A. x  e.  NN0  ( F `
 x )  C_  ( F `  ( x  +  1 ) ) )  ->  ( F `  b )  C_  ( F `  ( b  +  1 ) ) ) )
2423ancomsd 440 . . . . . 6  |-  ( b  e.  ( ZZ>= `  A
)  ->  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0 )  ->  ( F `  b )  C_  ( F `  (
b  +  1 ) ) ) )
25 sstr2 3186 . . . . . . 7  |-  ( ( F `  A ) 
C_  ( F `  b )  ->  (
( F `  b
)  C_  ( F `  ( b  +  1 ) )  ->  ( F `  A )  C_  ( F `  (
b  +  1 ) ) ) )
2625com12 27 . . . . . 6  |-  ( ( F `  b ) 
C_  ( F `  ( b  +  1 ) )  ->  (
( F `  A
)  C_  ( F `  b )  ->  ( F `  A )  C_  ( F `  (
b  +  1 ) ) ) )
2724, 26syl6 29 . . . . 5  |-  ( b  e.  ( ZZ>= `  A
)  ->  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0 )  ->  (
( F `  A
)  C_  ( F `  b )  ->  ( F `  A )  C_  ( F `  (
b  +  1 ) ) ) ) )
2827a2d 23 . . . 4  |-  ( b  e.  ( ZZ>= `  A
)  ->  ( (
( A. x  e. 
NN0  ( F `  x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  b )
)  ->  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  (
b  +  1 ) ) ) ) )
293, 6, 9, 12, 14, 28uzind4 10276 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  B
) ) )
3029com12 27 . 2  |-  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0 )  ->  ( B  e.  ( ZZ>= `  A )  ->  ( F `  A )  C_  ( F `  B
) ) )
31303impia 1148 1  |-  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0  /\  B  e.  ( ZZ>= `  A )
)  ->  ( F `  A )  C_  ( F `  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230
This theorem is referenced by:  nacsfix  26787
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-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  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-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231
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