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Theorem incssnn0 30539
Description: Transitivity induction of subsets, lemma for nacsfix 30540. (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 5871 . . . . . 6  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
21sseq2d 3537 . . . . 5  |-  ( a  =  A  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  A )
) )
32imbi2d 316 . . . 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 5871 . . . . . 6  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
54sseq2d 3537 . . . . 5  |-  ( a  =  b  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  b )
) )
65imbi2d 316 . . . 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 5871 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( F `  a )  =  ( F `  ( b  +  1 ) ) )
87sseq2d 3537 . . . . 5  |-  ( a  =  ( b  +  1 )  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  ( b  +  1 ) ) ) )
98imbi2d 316 . . . 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 5871 . . . . . 6  |-  ( a  =  B  ->  ( F `  a )  =  ( F `  B ) )
1110sseq2d 3537 . . . . 5  |-  ( a  =  B  ->  (
( F `  A
)  C_  ( F `  a )  <->  ( F `  A )  C_  ( F `  B )
) )
1211imbi2d 316 . . . 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 3528 . . . . 5  |-  ( F `
 A )  C_  ( F `  A )
1413a1ii 27 . . . 4  |-  ( A  e.  ZZ  ->  (
( A. x  e. 
NN0  ( F `  x )  C_  ( F `  ( x  +  1 ) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  A )
) )
15 eluznn0 11161 . . . . . . . . . 10  |-  ( ( A  e.  NN0  /\  b  e.  ( ZZ>= `  A ) )  -> 
b  e.  NN0 )
1615ancoms 453 . . . . . . . . 9  |-  ( ( b  e.  ( ZZ>= `  A )  /\  A  e.  NN0 )  ->  b  e.  NN0 )
17 fveq2 5871 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( F `  x )  =  ( F `  b ) )
18 oveq1 6301 . . . . . . . . . . . 12  |-  ( x  =  b  ->  (
x  +  1 )  =  ( b  +  1 ) )
1918fveq2d 5875 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( F `  ( x  +  1 ) )  =  ( F `  ( b  +  1 ) ) )
2017, 19sseq12d 3538 . . . . . . . . . 10  |-  ( x  =  b  ->  (
( F `  x
)  C_  ( F `  ( x  +  1 ) )  <->  ( F `  b )  C_  ( F `  ( b  +  1 ) ) ) )
2120rspcv 3215 . . . . . . . . 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 603 . . . . . . 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 454 . . . . . 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 3516 . . . . . . 7  |-  ( ( F `  A ) 
C_  ( F `  b )  ->  (
( F `  b
)  C_  ( F `  ( b  +  1 ) )  ->  ( F `  A )  C_  ( F `  (
b  +  1 ) ) ) )
2625com12 31 . . . . . 6  |-  ( ( F `  b ) 
C_  ( F `  ( b  +  1 ) )  ->  (
( F `  A
)  C_  ( F `  b )  ->  ( F `  A )  C_  ( F `  (
b  +  1 ) ) ) )
2724, 26syl6 33 . . . . 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 26 . . . 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 11149 . . 3  |-  ( B  e.  ( ZZ>= `  A
)  ->  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0 )  ->  ( F `  A )  C_  ( F `  B
) ) )
3029com12 31 . 2  |-  ( ( A. x  e.  NN0  ( F `  x ) 
C_  ( F `  ( x  +  1
) )  /\  A  e.  NN0 )  ->  ( B  e.  ( ZZ>= `  A )  ->  ( F `  A )  C_  ( F `  B
) ) )
31303impia 1193 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817    C_ wss 3481   ` cfv 5593  (class class class)co 6294   1c1 9503    + caddc 9505   NN0cn0 10805   ZZcz 10874   ZZ>=cuz 11092
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 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
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-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-recs 7052  df-rdg 7086  df-er 7321  df-en 7527  df-dom 7528  df-sdom 7529  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-n0 10806  df-z 10875  df-uz 11093
This theorem is referenced by:  nacsfix  30540
  Copyright terms: Public domain W3C validator