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Theorem funcnv4mpt 27184
Description: Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)
Hypotheses
Ref Expression
funcnvmpt.0  |-  F/ x ph
funcnvmpt.1  |-  F/_ x A
funcnvmpt.2  |-  F/_ x F
funcnvmpt.3  |-  F  =  ( x  e.  A  |->  B )
funcnvmpt.4  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
Assertion
Ref Expression
funcnv4mpt  |-  ( ph  ->  ( Fun  `' F  <->  A. i  e.  A  A. j  e.  A  (
i  =  j  \/ 
[_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B ) ) )
Distinct variable groups:    i, j, x    A, i, j    B, i, j    i, F    x, V    ph, i, j
Allowed substitution hints:    ph( x)    A( x)    B( x)    F( x, j)    V( i, j)

Proof of Theorem funcnv4mpt
StepHypRef Expression
1 nfv 1683 . 2  |-  F/ i
ph
2 nfcv 2629 . 2  |-  F/_ i A
3 nfcv 2629 . 2  |-  F/_ i F
4 funcnvmpt.3 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 funcnvmpt.1 . . . 4  |-  F/_ x A
6 nfcv 2629 . . . 4  |-  F/_ i B
7 nfcsb1v 3451 . . . 4  |-  F/_ x [_ i  /  x ]_ B
8 csbeq1a 3444 . . . 4  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
95, 2, 6, 7, 8cbvmptf 27166 . . 3  |-  ( x  e.  A  |->  B )  =  ( i  e.  A  |->  [_ i  /  x ]_ B )
104, 9eqtri 2496 . 2  |-  F  =  ( i  e.  A  |-> 
[_ i  /  x ]_ B )
11 funcnvmpt.4 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
1211sbimi 1717 . . 3  |-  ( [ i  /  x ]
( ph  /\  x  e.  A )  ->  [ i  /  x ] B  e.  V )
13 funcnvmpt.0 . . . . 5  |-  F/ x ph
14 nfcv 2629 . . . . . 6  |-  F/_ x
i
1514, 5nfel 2642 . . . . 5  |-  F/ x  i  e.  A
1613, 15nfan 1875 . . . 4  |-  F/ x
( ph  /\  i  e.  A )
17 eleq1 2539 . . . . 5  |-  ( x  =  i  ->  (
x  e.  A  <->  i  e.  A ) )
1817anbi2d 703 . . . 4  |-  ( x  =  i  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  i  e.  A ) ) )
1916, 18sbie 2123 . . 3  |-  ( [ i  /  x ]
( ph  /\  x  e.  A )  <->  ( ph  /\  i  e.  A ) )
20 nfcv 2629 . . . . 5  |-  F/_ x V
217, 20nfel 2642 . . . 4  |-  F/ x [_ i  /  x ]_ B  e.  V
228eleq1d 2536 . . . 4  |-  ( x  =  i  ->  ( B  e.  V  <->  [_ i  /  x ]_ B  e.  V
) )
2321, 22sbie 2123 . . 3  |-  ( [ i  /  x ] B  e.  V  <->  [_ i  /  x ]_ B  e.  V
)
2412, 19, 233imtr3i 265 . 2  |-  ( (
ph  /\  i  e.  A )  ->  [_ i  /  x ]_ B  e.  V )
25 csbeq1 3438 . 2  |-  ( i  =  j  ->  [_ i  /  x ]_ B  = 
[_ j  /  x ]_ B )
261, 2, 3, 10, 24, 25funcnv5mpt 27183 1  |-  ( ph  ->  ( Fun  `' F  <->  A. i  e.  A  A. j  e.  A  (
i  =  j  \/ 
[_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379   F/wnf 1599   [wsb 1711    e. wcel 1767   F/_wnfc 2615    =/= wne 2662   A.wral 2814   [_csb 3435    |-> cmpt 4505   `'ccnv 4998   Fun wfun 5580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594
This theorem is referenced by:  disjdsct  27193
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