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Theorem mapfzcons 34990
Description: Extending a one-based mapping by adding a tuple at the end results in another mapping. (Contributed by Stefan O'Rear, 10-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
Hypothesis
Ref Expression
mapfzcons.1  |-  M  =  ( N  +  1 )
Assertion
Ref Expression
mapfzcons  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  ( A  u.  { <. M ,  C >. } )  e.  ( B  ^m  (
1 ... M ) ) )

Proof of Theorem mapfzcons
StepHypRef Expression
1 simp2 998 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  A  e.  ( B  ^m  (
1 ... N ) ) )
2 elmapex 7476 . . . . . . . . 9  |-  ( A  e.  ( B  ^m  ( 1 ... N
) )  ->  ( B  e.  _V  /\  (
1 ... N )  e. 
_V ) )
32simpld 457 . . . . . . . 8  |-  ( A  e.  ( B  ^m  ( 1 ... N
) )  ->  B  e.  _V )
433ad2ant2 1019 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  B  e.  _V )
5 ovex 6305 . . . . . . 7  |-  ( 1 ... N )  e. 
_V
6 elmapg 7469 . . . . . . 7  |-  ( ( B  e.  _V  /\  ( 1 ... N
)  e.  _V )  ->  ( A  e.  ( B  ^m  ( 1 ... N ) )  <-> 
A : ( 1 ... N ) --> B ) )
74, 5, 6sylancl 660 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  ( A  e.  ( B  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> B ) )
81, 7mpbid 210 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  A : ( 1 ... N ) --> B )
9 ovex 6305 . . . . . . . 8  |-  ( N  +  1 )  e. 
_V
10 simp3 999 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  C  e.  B )
11 f1osng 5836 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  _V  /\  C  e.  B )  ->  { <. ( N  + 
1 ) ,  C >. } : { ( N  +  1 ) } -1-1-onto-> { C } )
129, 10, 11sylancr 661 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  { <. ( N  +  1 ) ,  C >. } : { ( N  + 
1 ) } -1-1-onto-> { C } )
13 f1of 5798 . . . . . . 7  |-  ( {
<. ( N  +  1 ) ,  C >. } : { ( N  +  1 ) } -1-1-onto-> { C }  ->  { <. ( N  +  1 ) ,  C >. } : { ( N  + 
1 ) } --> { C } )
1412, 13syl 17 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  { <. ( N  +  1 ) ,  C >. } : { ( N  + 
1 ) } --> { C } )
15 snssi 4115 . . . . . . 7  |-  ( C  e.  B  ->  { C }  C_  B )
16153ad2ant3 1020 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  { C }  C_  B )
1714, 16fssd 5722 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  { <. ( N  +  1 ) ,  C >. } : { ( N  + 
1 ) } --> B )
18 fzp1disj 11791 . . . . . 6  |-  ( ( 1 ... N )  i^i  { ( N  +  1 ) } )  =  (/)
1918a1i 11 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  (
( 1 ... N
)  i^i  { ( N  +  1 ) } )  =  (/) )
20 fun 5730 . . . . 5  |-  ( ( ( A : ( 1 ... N ) --> B  /\  { <. ( N  +  1 ) ,  C >. } : { ( N  + 
1 ) } --> B )  /\  ( ( 1 ... N )  i^i 
{ ( N  + 
1 ) } )  =  (/) )  ->  ( A  u.  { <. ( N  +  1 ) ,  C >. } ) : ( ( 1 ... N )  u. 
{ ( N  + 
1 ) } ) --> ( B  u.  B
) )
218, 17, 19, 20syl21anc 1229 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  ( A  u.  { <. ( N  +  1 ) ,  C >. } ) : ( ( 1 ... N )  u. 
{ ( N  + 
1 ) } ) --> ( B  u.  B
) )
22 1z 10934 . . . . . . 7  |-  1  e.  ZZ
23 simp1 997 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  N  e.  NN0 )
24 nn0uz 11160 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
25 1m1e0 10644 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2625fveq2i 5851 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
2724, 26eqtr4i 2434 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
2823, 27syl6eleq 2500 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )
29 fzsuc2 11790 . . . . . . 7  |-  ( ( 1  e.  ZZ  /\  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... ( N  +  1 ) )  =  ( ( 1 ... N )  u.  { ( N  +  1 ) } ) )
3022, 28, 29sylancr 661 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  (
1 ... ( N  + 
1 ) )  =  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )
3130eqcomd 2410 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  (
( 1 ... N
)  u.  { ( N  +  1 ) } )  =  ( 1 ... ( N  +  1 ) ) )
32 unidm 3585 . . . . . 6  |-  ( B  u.  B )  =  B
3332a1i 11 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  ( B  u.  B )  =  B )
3431, 33feq23d 5708 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  (
( A  u.  { <. ( N  +  1 ) ,  C >. } ) : ( ( 1 ... N )  u.  { ( N  +  1 ) } ) --> ( B  u.  B )  <->  ( A  u.  { <. ( N  + 
1 ) ,  C >. } ) : ( 1 ... ( N  +  1 ) ) --> B ) )
3521, 34mpbid 210 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  ( A  u.  { <. ( N  +  1 ) ,  C >. } ) : ( 1 ... ( N  +  1 ) ) --> B )
36 ovex 6305 . . . 4  |-  ( 1 ... ( N  + 
1 ) )  e. 
_V
37 elmapg 7469 . . . 4  |-  ( ( B  e.  _V  /\  ( 1 ... ( N  +  1 ) )  e.  _V )  ->  ( ( A  u.  {
<. ( N  +  1 ) ,  C >. } )  e.  ( B  ^m  ( 1 ... ( N  +  1 ) ) )  <->  ( A  u.  { <. ( N  + 
1 ) ,  C >. } ) : ( 1 ... ( N  +  1 ) ) --> B ) )
384, 36, 37sylancl 660 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  (
( A  u.  { <. ( N  +  1 ) ,  C >. } )  e.  ( B  ^m  ( 1 ... ( N  +  1 ) ) )  <->  ( A  u.  { <. ( N  + 
1 ) ,  C >. } ) : ( 1 ... ( N  +  1 ) ) --> B ) )
3935, 38mpbird 232 . 2  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  ( A  u.  { <. ( N  +  1 ) ,  C >. } )  e.  ( B  ^m  ( 1 ... ( N  +  1 ) ) ) )
40 mapfzcons.1 . . . . 5  |-  M  =  ( N  +  1 )
4140opeq1i 4161 . . . 4  |-  <. M ,  C >.  =  <. ( N  +  1 ) ,  C >.
4241sneqi 3982 . . 3  |-  { <. M ,  C >. }  =  { <. ( N  + 
1 ) ,  C >. }
4342uneq2i 3593 . 2  |-  ( A  u.  { <. M ,  C >. } )  =  ( A  u.  { <. ( N  +  1 ) ,  C >. } )
4440oveq2i 6288 . . 3  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
4544oveq2i 6288 . 2  |-  ( B  ^m  ( 1 ... M ) )  =  ( B  ^m  (
1 ... ( N  + 
1 ) ) )
4639, 43, 453eltr4g 2508 1  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  ( A  u.  { <. M ,  C >. } )  e.  ( B  ^m  (
1 ... M ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 974    = wceq 1405    e. wcel 1842   _Vcvv 3058    u. cun 3411    i^i cin 3412    C_ wss 3413   (/)c0 3737   {csn 3971   <.cop 3977   -->wf 5564   -1-1-onto->wf1o 5567   ` cfv 5568  (class class class)co 6277    ^m cmap 7456   0cc0 9521   1c1 9522    + caddc 9524    - cmin 9840   NN0cn0 10835   ZZcz 10904   ZZ>=cuz 11126   ...cfz 11724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725
This theorem is referenced by:  rexrabdioph  35069
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