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Theorem incssnn0 26766
Description: Transitivity induction of subsets, lemma for nacsfix 26767. (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 5729 . . . . . 6  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
21sseq2d 3377 . . . . 5  |-  ( a  =  A  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  A )
) )
32imbi2d 309 . . . 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 5729 . . . . . 6  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
54sseq2d 3377 . . . . 5  |-  ( a  =  b  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  b )
) )
65imbi2d 309 . . . 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 5729 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( F `  a )  =  ( F `  ( b  +  1 ) ) )
87sseq2d 3377 . . . . 5  |-  ( a  =  ( b  +  1 )  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  ( b  +  1 ) ) ) )
98imbi2d 309 . . . 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 5729 . . . . . 6  |-  ( a  =  B  ->  ( F `  a )  =  ( F `  B ) )
1110sseq2d 3377 . . . . 5  |-  ( a  =  B  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  B )
) )
1211imbi2d 309 . . . 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 3368 . . . . 5  |-  ( F `
 A )  C_  ( F `  A )
1413a1ii 26 . . . 4  |-  ( A  e.  ZZ  ->  (
( A. x  e. 
NN0  ( F `  x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  A )
) )
15 eluznn0 10547 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  b  e.  ( ZZ>= `  A ) )  -> 
b  e.  NN0 )
1615ancoms 441 . . . . . . . . 9  |-  ( ( b  e.  ( ZZ>= `  A )  /\  A  e.  NN0 )  ->  b  e.  NN0 )
17 fveq2 5729 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( F `  x )  =  ( F `  b ) )
18 oveq1 6089 . . . . . . . . . . . 12  |-  ( x  =  b  ->  (
x  +  1 )  =  ( b  +  1 ) )
1918fveq2d 5733 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( F `  ( x  +  1 ) )  =  ( F `  ( b  +  1 ) ) )
2017, 19sseq12d 3378 . . . . . . . . . 10  |-  ( x  =  b  ->  (
( F `  x
)  C_  ( F `  ( x  +  1 ) )  <->  ( F `  b )  C_  ( F `  ( b  +  1 ) ) ) )
2120rspcv 3049 . . . . . . . . 9  |-  ( b  e.  NN0  ->  ( A. x  e.  NN0  ( F `
 x )  C_  ( F `  ( x  +  1 ) )  ->  ( F `  b )  C_  ( F `  ( b  +  1 ) ) ) )
2216, 21syl 16 . . . . . . . 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 588 . . . . . . 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 442 . . . . . 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 3356 . . . . . . 7  |-  ( ( F `  A ) 
C_  ( F `  b )  ->  (
( F `  b
)  C_  ( F `  ( b  +  1 ) )  ->  ( F `  A )  C_  ( F `  (
b  +  1 ) ) ) )
2625com12 30 . . . . . 6  |-  ( ( F `  b ) 
C_  ( F `  ( b  +  1 ) )  ->  (
( F `  A
)  C_  ( F `  b )  ->  ( F `  A )  C_  ( F `  (
b  +  1 ) ) ) )
2724, 26syl6 32 . . . . 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 25 . . . 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 10535 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  B
) ) )
3029com12 30 . 2  |-  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0 )  ->  ( B  e.  ( ZZ>= `  A )  ->  ( F `  A )  C_  ( F `  B
) ) )
31303impia 1151 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2706    C_ wss 3321   ` cfv 5455  (class class class)co 6082   1c1 8992    + caddc 8994   NN0cn0 10222   ZZcz 10283   ZZ>=cuz 10489
This theorem is referenced by:  nacsfix  26767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-n0 10223  df-z 10284  df-uz 10490
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