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Theorem mapfzcons 30624
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 7441 . . . . . . . . 9  |-  ( A  e.  ( B  ^m  ( 1 ... N
) )  ->  ( B  e.  _V  /\  (
1 ... N )  e. 
_V ) )
32simpld 459 . . . . . . . 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 6309 . . . . . . 7  |-  ( 1 ... N )  e. 
_V
6 elmapg 7435 . . . . . . 7  |-  ( ( B  e.  _V  /\  ( 1 ... N
)  e.  _V )  ->  ( A  e.  ( B  ^m  ( 1 ... N ) )  <-> 
A : ( 1 ... N ) --> B ) )
74, 5, 6sylancl 662 . . . . . 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 6309 . . . . . . . 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 5844 . . . . . . . 8  |-  ( ( ( N  +  1 )  e.  _V  /\  C  e.  B )  ->  { <. ( N  + 
1 ) ,  C >. } : { ( N  +  1 ) } -1-1-onto-> { C } )
129, 10, 11sylancr 663 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  { <. ( N  +  1 ) ,  C >. } : { ( N  + 
1 ) } -1-1-onto-> { C } )
13 f1of 5806 . . . . . . 7  |-  ( {
<. ( N  +  1 ) ,  C >. } : { ( N  +  1 ) } -1-1-onto-> { C }  ->  { <. ( N  +  1 ) ,  C >. } : { ( N  + 
1 ) } --> { C } )
1412, 13syl 16 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  { <. ( N  +  1 ) ,  C >. } : { ( N  + 
1 ) } --> { C } )
15 snssi 4159 . . . . . . 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 5730 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  { <. ( N  +  1 ) ,  C >. } : { ( N  + 
1 ) } --> B )
18 fzp1disj 11749 . . . . . 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 5738 . . . . 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 1228 . . . 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 10901 . . . . . . 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 11126 . . . . . . . . 9  |-  NN0  =  ( ZZ>= `  0 )
25 1m1e0 10611 . . . . . . . . . 10  |-  ( 1  -  1 )  =  0
2625fveq2i 5859 . . . . . . . . 9  |-  ( ZZ>= `  ( 1  -  1 ) )  =  (
ZZ>= `  0 )
2724, 26eqtr4i 2475 . . . . . . . 8  |-  NN0  =  ( ZZ>= `  ( 1  -  1 ) )
2823, 27syl6eleq 2541 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )
29 fzsuc2 11748 . . . . . . 7  |-  ( ( 1  e.  ZZ  /\  N  e.  ( ZZ>= `  ( 1  -  1 ) ) )  -> 
( 1 ... ( N  +  1 ) )  =  ( ( 1 ... N )  u.  { ( N  +  1 ) } ) )
3022, 28, 29sylancr 663 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  (
1 ... ( N  + 
1 ) )  =  ( ( 1 ... N )  u.  {
( N  +  1 ) } ) )
3130eqcomd 2451 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ( B  ^m  ( 1 ... N
) )  /\  C  e.  B )  ->  (
( 1 ... N
)  u.  { ( N  +  1 ) } )  =  ( 1 ... ( N  +  1 ) ) )
32 unidm 3632 . . . . . 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 5716 . . . 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 6309 . . . 4  |-  ( 1 ... ( N  + 
1 ) )  e. 
_V
37 elmapg 7435 . . . 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 662 . . 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 4205 . . . 4  |-  <. M ,  C >.  =  <. ( N  +  1 ) ,  C >.
4241sneqi 4025 . . 3  |-  { <. M ,  C >. }  =  { <. ( N  + 
1 ) ,  C >. }
4342uneq2i 3640 . 2  |-  ( A  u.  { <. M ,  C >. } )  =  ( A  u.  { <. ( N  +  1 ) ,  C >. } )
4440oveq2i 6292 . . 3  |-  ( 1 ... M )  =  ( 1 ... ( N  +  1 ) )
4544oveq2i 6292 . 2  |-  ( B  ^m  ( 1 ... M ) )  =  ( B  ^m  (
1 ... ( N  + 
1 ) ) )
4639, 43, 453eltr4g 2549 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 1383    e. wcel 1804   _Vcvv 3095    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3770   {csn 4014   <.cop 4020   -->wf 5574   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281    ^m cmap 7422   0cc0 9495   1c1 9496    + caddc 9498    - cmin 9810   NN0cn0 10802   ZZcz 10871   ZZ>=cuz 11092   ...cfz 11683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684
This theorem is referenced by:  rexrabdioph  30703
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