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Theorem fnctartar 26010
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 6801 . . . . 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 5416 . . . . . . . . . . 11  |-  ( f : A --> T  -> 
f  C_  ( A  X.  T ) )
4 ssel 3187 . . . . . . . . . . . . 13  |-  ( f 
C_  ( A  X.  T )  ->  (
x  e.  f  ->  x  e.  ( A  X.  T ) ) )
5 elxp6 6167 . . . . . . . . . . . . . 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 4136 . . . . . . . . . . . . . . . . . . . . . . . . 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 8409 . . . . . . . . . . . . . . . . . 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 2356 . . . . . . . . . . . . . . . . . 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 3198 . . . . . . . 8  |-  ( ( f : A --> T  /\  ( T  e.  Tarski  /\  Tr  T  /\  A  e.  T
) )  ->  f  C_  T )
30 ffn 5405 . . . . . . . . . 10  |-  ( f : A --> T  -> 
f  Fn  A )
31 fnfun 5357 . . . . . . . . . . 11  |-  ( f  Fn  A  ->  Fun  f )
32 fndm 5359 . . . . . . . . . . 11  |-  ( f  Fn  A  ->  dom  f  =  A )
3331, 32jca 518 . . . . . . . . . 10  |-  ( f  Fn  A  ->  ( Fun  f  /\  dom  f  =  A ) )
34 vex 2804 . . . . . . . . . . . 12  |-  f  e. 
_V
3534fundmen 6950 . . . . . . . . . . 11  |-  ( Fun  f  ->  dom  f  ~~  f )
36 breq1 4042 . . . . . . . . . . . 12  |-  ( dom  f  =  A  -> 
( dom  f  ~~  f 
<->  A  ~~  f ) )
37 tsksdom 8394 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  Tarski  /\  A  e.  T )  ->  A  ~<  T )
38 sdomen1 7021 . . . . . . . . . . . . . . 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 8395 . . . . . 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 3198 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 1632    e. wcel 1696    C_ wss 3165   <.cop 3656   class class class wbr 4039   Tr wtr 4129    X. cxp 4703   dom cdm 4705   Fun wfun 5265    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137    ^m cmap 6788    ~~ cen 6876    ~< csdm 6878   Tarskictsk 8386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-r1 7452  df-tsk 8387
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