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Theorem incssnn0 30611
Description: Transitivity induction of subsets, lemma for nacsfix 30612. (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 5852 . . . . . 6  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
21sseq2d 3514 . . . . 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 5852 . . . . . 6  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
54sseq2d 3514 . . . . 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 5852 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( F `  a )  =  ( F `  ( b  +  1 ) ) )
87sseq2d 3514 . . . . 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 5852 . . . . . 6  |-  ( a  =  B  ->  ( F `  a )  =  ( F `  B ) )
1110sseq2d 3514 . . . . 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 3505 . . . . 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 11155 . . . . . . . . . 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 5852 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( F `  x )  =  ( F `  b ) )
18 oveq1 6284 . . . . . . . . . . . 12  |-  ( x  =  b  ->  (
x  +  1 )  =  ( b  +  1 ) )
1918fveq2d 5856 . . . . . . . . . . 11  |-  ( x  =  b  ->  ( F `  ( x  +  1 ) )  =  ( F `  ( b  +  1 ) ) )
2017, 19sseq12d 3515 . . . . . . . . . 10  |-  ( x  =  b  ->  (
( F `  x
)  C_  ( F `  ( x  +  1 ) )  <->  ( F `  b )  C_  ( F `  ( b  +  1 ) ) ) )
2120rspcv 3190 . . . . . . . . 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 3493 . . . . . . 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 11143 . . 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 1192 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 972    = wceq 1381    e. wcel 1802   A.wral 2791    C_ wss 3458   ` cfv 5574  (class class class)co 6277   1c1 9491    + caddc 9493   NN0cn0 10796   ZZcz 10865   ZZ>=cuz 11085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-n0 10797  df-z 10866  df-uz 11086
This theorem is referenced by:  nacsfix  30612
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