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Theorem fnctartar 25907
Description: Consider functions whose domain  A is an element of a transitive Tarski's class  T and whose range is  T, then they are elements of  T. CLASSES2 th. 23. (Contributed by FL, 26-Sep-2011.) (Revised by Mario Carneiro, 4-May-2015.)
Assertion
Ref Expression
fnctartar  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  ( T  ^m  A )  C_  T
)

Proof of Theorem fnctartar
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapg 6785 . . . . 5  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  (
f  e.  ( T  ^m  A )  <->  f : A
--> T ) )
2 simpr1 961 . . . . . . . 8  |-  ( ( f : A --> T  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  T  e.  Tarski )
3 fssxp 5400 . . . . . . . . . . 11  |-  ( f : A --> T  -> 
f  C_  ( A  X.  T ) )
4 ssel 3174 . . . . . . . . . . . . 13  |-  ( f 
C_  ( A  X.  T )  ->  (
x  e.  f  ->  x  e.  ( A  X.  T ) ) )
5 elxp6 6151 . . . . . . . . . . . . . 14  |-  ( x  e.  ( A  X.  T )  <->  ( x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  /\  (
( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  T ) ) )
6 simp31 991 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  T )  /\  f : A --> T  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  T  e.  Tarski )
7 trel 4120 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( Tr  T  ->  ( (
( 1st `  x
)  e.  A  /\  A  e.  T )  ->  ( 1st `  x
)  e.  T ) )
87exp3acom23 1362 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( Tr  T  ->  ( A  e.  T  ->  ( ( 1st `  x )  e.  A  ->  ( 1st `  x )  e.  T ) ) )
98imp 418 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( Tr  T  /\  A  e.  T )  ->  (
( 1st `  x
)  e.  A  -> 
( 1st `  x
)  e.  T ) )
1093adant1 973 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  ( ( 1st `  x )  e.  A  ->  ( 1st `  x )  e.  T
) )
1110com12 27 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1st `  x )  e.  A  ->  (
( T  e.  Tarski  /\ 
Tr  T  /\  A  e.  T )  ->  ( 1st `  x )  e.  T ) )
1211adantr 451 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  T )  -> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  ( 1st `  x
)  e.  T ) )
1312imp 418 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  T )  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  ( 1st `  x )  e.  T )
14133adant2 974 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  T )  /\  f : A --> T  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  ( 1st `  x )  e.  T )
15 simp1r 980 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  T )  /\  f : A --> T  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  ( 2nd `  x )  e.  T )
16 tskop 8393 . . . . . . . . . . . . . . . . . 18  |-  ( ( T  e.  Tarski  /\  ( 1st `  x )  e.  T  /\  ( 2nd `  x )  e.  T
)  ->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  T )
176, 14, 15, 16syl3anc 1182 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  T )  /\  f : A --> T  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  T )
18173exp 1150 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  T )  -> 
( f : A --> T  ->  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  T ) ) )
19 eleq1 2343 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  ->  (
x  e.  T  <->  <. ( 1st `  x ) ,  ( 2nd `  x )
>.  e.  T ) )
2019imbi2d 307 . . . . . . . . . . . . . . . . 17  |-  ( x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  ->  (
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  x  e.  T )  <-> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  -> 
<. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  T
) ) )
2120imbi2d 307 . . . . . . . . . . . . . . . 16  |-  ( x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  ->  (
( f : A --> T  ->  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  x  e.  T ) )  <->  ( f : A --> T  ->  (
( T  e.  Tarski  /\ 
Tr  T  /\  A  e.  T )  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  T ) ) ) )
2218, 21syl5ibr 212 . . . . . . . . . . . . . . 15  |-  ( x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  ->  (
( ( 1st `  x
)  e.  A  /\  ( 2nd `  x )  e.  T )  -> 
( f : A --> T  ->  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  x  e.  T ) ) ) )
2322imp 418 . . . . . . . . . . . . . 14  |-  ( ( x  =  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  /\  ( ( 1st `  x )  e.  A  /\  ( 2nd `  x )  e.  T
) )  ->  (
f : A --> T  -> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  x  e.  T ) ) )
245, 23sylbi 187 . . . . . . . . . . . . 13  |-  ( x  e.  ( A  X.  T )  ->  (
f : A --> T  -> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  x  e.  T ) ) )
254, 24syl6com 31 . . . . . . . . . . . 12  |-  ( x  e.  f  ->  (
f  C_  ( A  X.  T )  ->  (
f : A --> T  -> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  x  e.  T ) ) ) )
2625com4l 78 . . . . . . . . . . 11  |-  ( f 
C_  ( A  X.  T )  ->  (
f : A --> T  -> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  ( x  e.  f  ->  x  e.  T
) ) ) )
273, 26mpcom 32 . . . . . . . . . 10  |-  ( f : A --> T  -> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  ( x  e.  f  ->  x  e.  T
) ) )
2827imp 418 . . . . . . . . 9  |-  ( ( f : A --> T  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  (
x  e.  f  ->  x  e.  T )
)
2928ssrdv 3185 . . . . . . . 8  |-  ( ( f : A --> T  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  f  C_  T )
30 ffn 5389 . . . . . . . . . 10  |-  ( f : A --> T  -> 
f  Fn  A )
31 fnfun 5341 . . . . . . . . . . 11  |-  ( f  Fn  A  ->  Fun  f )
32 fndm 5343 . . . . . . . . . . 11  |-  ( f  Fn  A  ->  dom  f  =  A )
3331, 32jca 518 . . . . . . . . . 10  |-  ( f  Fn  A  ->  ( Fun  f  /\  dom  f  =  A ) )
34 vex 2791 . . . . . . . . . . . 12  |-  f  e. 
_V
3534fundmen 6934 . . . . . . . . . . 11  |-  ( Fun  f  ->  dom  f  ~~  f )
36 breq1 4026 . . . . . . . . . . . 12  |-  ( dom  f  =  A  -> 
( dom  f  ~~  f 
<->  A  ~~  f ) )
37 tsksdom 8378 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )
38 sdomen1 7005 . . . . . . . . . . . . . . 15  |-  ( A 
~~  f  ->  ( A  ~<  T  <->  f  ~<  T ) )
3937, 38syl5ibcom 211 . . . . . . . . . . . . . 14  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  ( A  ~~  f  ->  f  ~<  T ) )
40393adant2 974 . . . . . . . . . . . . 13  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  ( A  ~~  f  ->  f  ~<  T ) )
4140com12 27 . . . . . . . . . . . 12  |-  ( A 
~~  f  ->  (
( T  e.  Tarski  /\ 
Tr  T  /\  A  e.  T )  ->  f  ~<  T ) )
4236, 41syl6bi 219 . . . . . . . . . . 11  |-  ( dom  f  =  A  -> 
( dom  f  ~~  f  ->  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  f  ~<  T ) ) )
4335, 42mpan9 455 . . . . . . . . . 10  |-  ( ( Fun  f  /\  dom  f  =  A )  ->  ( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  f  ~<  T )
)
4430, 33, 433syl 18 . . . . . . . . 9  |-  ( f : A --> T  -> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  f  ~<  T )
)
4544imp 418 . . . . . . . 8  |-  ( ( f : A --> T  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  f  ~<  T )
462, 29, 453jca 1132 . . . . . . 7  |-  ( ( f : A --> T  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  ( T  e.  Tarski  /\  f  C_  T  /\  f  ~<  T ) )
4746ex 423 . . . . . 6  |-  ( f : A --> T  -> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  ( T  e.  Tarski  /\  f  C_  T  /\  f  ~<  T ) ) )
48 tskssel 8379 . . . . . 6  |-  ( ( T  e.  Tarski  /\  f  C_  T  /\  f  ~<  T )  ->  f  e.  T )
4947, 48syl6 29 . . . . 5  |-  ( f : A --> T  -> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  f  e.  T ) )
501, 49syl6bi 219 . . . 4  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  (
f  e.  ( T  ^m  A )  -> 
( ( T  e. 
Tarski  /\  Tr  T  /\  A  e.  T )  ->  f  e.  T ) ) )
51503adant2 974 . . 3  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  ( f  e.  ( T  ^m  A
)  ->  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  f  e.  T ) ) )
5251pm2.43a 45 . 2  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  ( f  e.  ( T  ^m  A
)  ->  f  e.  T ) )
5352ssrdv 3185 1  |-  ( ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
)  ->  ( T  ^m  A )  C_  T
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   <.cop 3643   class class class wbr 4023   Tr wtr 4113    X. cxp 4687   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772    ~~ cen 6860    ~< csdm 6862   Tarskictsk 8370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-r1 7436  df-tsk 8371
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