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Theorem funcnv4mpt 27941
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 1728 . 2  |-  F/ i
ph
2 nfcv 2564 . 2  |-  F/_ i A
3 nfcv 2564 . 2  |-  F/_ i F
4 funcnvmpt.3 . . 3  |-  F  =  ( x  e.  A  |->  B )
5 funcnvmpt.1 . . . 4  |-  F/_ x A
6 nfcv 2564 . . . 4  |-  F/_ i B
7 nfcsb1v 3388 . . . 4  |-  F/_ x [_ i  /  x ]_ B
8 csbeq1a 3381 . . . 4  |-  ( x  =  i  ->  B  =  [_ i  /  x ]_ B )
95, 2, 6, 7, 8cbvmptf 4484 . . 3  |-  ( x  e.  A  |->  B )  =  ( i  e.  A  |->  [_ i  /  x ]_ B )
104, 9eqtri 2431 . 2  |-  F  =  ( i  e.  A  |-> 
[_ i  /  x ]_ B )
11 funcnvmpt.4 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
1211sbimi 1769 . . 3  |-  ( [ i  /  x ]
( ph  /\  x  e.  A )  ->  [ i  /  x ] B  e.  V )
13 funcnvmpt.0 . . . . 5  |-  F/ x ph
14 nfcv 2564 . . . . . 6  |-  F/_ x
i
1514, 5nfel 2577 . . . . 5  |-  F/ x  i  e.  A
1613, 15nfan 1956 . . . 4  |-  F/ x
( ph  /\  i  e.  A )
17 eleq1 2474 . . . . 5  |-  ( x  =  i  ->  (
x  e.  A  <->  i  e.  A ) )
1817anbi2d 702 . . . 4  |-  ( x  =  i  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  i  e.  A ) ) )
1916, 18sbie 2173 . . 3  |-  ( [ i  /  x ]
( ph  /\  x  e.  A )  <->  ( ph  /\  i  e.  A ) )
20 nfcv 2564 . . . . 5  |-  F/_ x V
217, 20nfel 2577 . . . 4  |-  F/ x [_ i  /  x ]_ B  e.  V
228eleq1d 2471 . . . 4  |-  ( x  =  i  ->  ( B  e.  V  <->  [_ i  /  x ]_ B  e.  V
) )
2321, 22sbie 2173 . . 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 3375 . 2  |-  ( i  =  j  ->  [_ i  /  x ]_ B  = 
[_ j  /  x ]_ B )
261, 2, 3, 10, 24, 25funcnv5mpt 27940 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 366    /\ wa 367    = wceq 1405   F/wnf 1637   [wsb 1763    e. wcel 1842   F/_wnfc 2550    =/= wne 2598   A.wral 2753   [_csb 3372    |-> cmpt 4452   `'ccnv 4821   Fun wfun 5562
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-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  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-ral 2758  df-rex 2759  df-rmo 2761  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-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  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-iota 5532  df-fun 5570  df-fn 5571  df-fv 5576
This theorem is referenced by:  disjdsct  27951
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