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Theorem funcnvmpt 25992
Description: Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-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
funcnvmpt  |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x  e.  A  y  =  B ) )
Distinct variable groups:    x, y    y, F    ph, y
Allowed substitution hints:    ph( x)    A( x, y)    B( x, y)    F( x)    V( x, y)

Proof of Theorem funcnvmpt
StepHypRef Expression
1 relcnv 5211 . . . 4  |-  Rel  `' F
2 nfcv 2584 . . . . 5  |-  F/_ y `' F
3 funcnvmpt.2 . . . . . 6  |-  F/_ x F
43nfcnv 5023 . . . . 5  |-  F/_ x `' F
52, 4dffun6f 5437 . . . 4  |-  ( Fun  `' F  <->  ( Rel  `' F  /\  A. y E* x  y `' F x ) )
61, 5mpbiran 909 . . 3  |-  ( Fun  `' F  <->  A. y E* x  y `' F x )
7 vex 2980 . . . . . 6  |-  y  e. 
_V
8 vex 2980 . . . . . 6  |-  x  e. 
_V
97, 8brcnv 5027 . . . . 5  |-  ( y `' F x  <->  x F
y )
109mobii 2279 . . . 4  |-  ( E* x  y `' F x 
<->  E* x  x F y )
1110albii 1610 . . 3  |-  ( A. y E* x  y `' F x  <->  A. y E* x  x F
y )
126, 11bitri 249 . 2  |-  ( Fun  `' F  <->  A. y E* x  x F y )
13 funcnvmpt.0 . . . . 5  |-  F/ x ph
14 funcnvmpt.3 . . . . . . . . . 10  |-  F  =  ( x  e.  A  |->  B )
1514funmpt2 5460 . . . . . . . . 9  |-  Fun  F
16 funbrfv2b 5741 . . . . . . . . 9  |-  ( Fun 
F  ->  ( x F y  <->  ( x  e.  dom  F  /\  ( F `  x )  =  y ) ) )
1715, 16ax-mp 5 . . . . . . . 8  |-  ( x F y  <->  ( x  e.  dom  F  /\  ( F `  x )  =  y ) )
18 funcnvmpt.4 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
19 elex 2986 . . . . . . . . . . . . . . 15  |-  ( B  e.  V  ->  B  e.  _V )
2018, 19syl 16 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  _V )
2120ex 434 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  A  ->  B  e.  _V )
)
2213, 21ralrimi 2802 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  A  B  e.  _V )
23 funcnvmpt.1 . . . . . . . . . . . . 13  |-  F/_ x A
2423rabid2f 25890 . . . . . . . . . . . 12  |-  ( A  =  { x  e.  A  |  B  e. 
_V }  <->  A. x  e.  A  B  e.  _V )
2522, 24sylibr 212 . . . . . . . . . . 11  |-  ( ph  ->  A  =  { x  e.  A  |  B  e.  _V } )
2614dmmpt 5338 . . . . . . . . . . 11  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
2725, 26syl6reqr 2494 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  A )
2827eleq2d 2510 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  dom  F  <-> 
x  e.  A ) )
2928anbi1d 704 . . . . . . . 8  |-  ( ph  ->  ( ( x  e. 
dom  F  /\  ( F `  x )  =  y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
3017, 29syl5bb 257 . . . . . . 7  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  ( F `  x
)  =  y ) ) )
3130bian1d 25856 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  x F y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
32 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
3314fveq1i 5697 . . . . . . . . . . 11  |-  ( F `
 x )  =  ( ( x  e.  A  |->  B ) `  x )
3423fvmpt2f 25980 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3533, 34syl5eq 2487 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( F `  x
)  =  B )
3632, 18, 35syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
3736eqeq2d 2454 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  y  =  B ) )
38 eqcom 2445 . . . . . . . . 9  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
3928biimpar 485 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  dom  F )
40 funbrfvb 5739 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  y  <-> 
x F y ) )
4115, 39, 40sylancr 663 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  =  y  <->  x F
y ) )
4238, 41syl5bbr 259 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
4337, 42bitr3d 255 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  B  <->  x F
y ) )
4443pm5.32da 641 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  x F y ) ) )
4531, 44, 303bitr4rd 286 . . . . 5  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  y  =  B
) ) )
4613, 45mobid 2275 . . . 4  |-  ( ph  ->  ( E* x  x F y  <->  E* x
( x  e.  A  /\  y  =  B
) ) )
47 df-rmo 2728 . . . 4  |-  ( E* x  e.  A  y  =  B  <->  E* x
( x  e.  A  /\  y  =  B
) )
4846, 47syl6bbr 263 . . 3  |-  ( ph  ->  ( E* x  x F y  <->  E* x  e.  A  y  =  B ) )
4948albidv 1679 . 2  |-  ( ph  ->  ( A. y E* x  x F y  <->  A. y E* x  e.  A  y  =  B ) )
5012, 49syl5bb 257 1  |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x  e.  A  y  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369   F/wnf 1589    e. wcel 1756   E*wmo 2254   F/_wnfc 2571   A.wral 2720   E*wrmo 2723   {crab 2724   _Vcvv 2977   class class class wbr 4297    e. cmpt 4355   `'ccnv 4844   dom cdm 4845   Rel wrel 4850   Fun wfun 5417   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431
This theorem is referenced by:  funcnv5mpt  25993
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