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Theorem funcnvmpt 27210
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 5374 . . . 4  |-  Rel  `' F
2 nfcv 2629 . . . . 5  |-  F/_ y `' F
3 funcnvmpt.2 . . . . . 6  |-  F/_ x F
43nfcnv 5181 . . . . 5  |-  F/_ x `' F
52, 4dffun6f 5602 . . . 4  |-  ( Fun  `' F  <->  ( Rel  `' F  /\  A. y E* x  y `' F x ) )
61, 5mpbiran 916 . . 3  |-  ( Fun  `' F  <->  A. y E* x  y `' F x )
7 vex 3116 . . . . . 6  |-  y  e. 
_V
8 vex 3116 . . . . . 6  |-  x  e. 
_V
97, 8brcnv 5185 . . . . 5  |-  ( y `' F x  <->  x F
y )
109mobii 2301 . . . 4  |-  ( E* x  y `' F x 
<->  E* x  x F y )
1110albii 1620 . . 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 5625 . . . . . . . . 9  |-  Fun  F
16 funbrfv2b 5912 . . . . . . . . 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 3122 . . . . . . . . . . . . . . 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 2864 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  A  B  e.  _V )
23 funcnvmpt.1 . . . . . . . . . . . . 13  |-  F/_ x A
2423rabid2f 27104 . . . . . . . . . . . 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 5502 . . . . . . . . . . 11  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
2725, 26syl6reqr 2527 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  A )
2827eleq2d 2537 . . . . . . . . 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 27070 . . . . . 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 5867 . . . . . . . . . . 11  |-  ( F `
 x )  =  ( ( x  e.  A  |->  B ) `  x )
3423fvmpt2f 27198 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3533, 34syl5eq 2520 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( F `  x
)  =  B )
3632, 18, 35syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
3736eqeq2d 2481 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  y  =  B ) )
38 eqcom 2476 . . . . . . . . 9  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
3928biimpar 485 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  dom  F )
40 funbrfvb 5910 . . . . . . . . . 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 2297 . . . 4  |-  ( ph  ->  ( E* x  x F y  <->  E* x
( x  e.  A  /\  y  =  B
) ) )
47 df-rmo 2822 . . . 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 1689 . 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 1377    = wceq 1379   F/wnf 1599    e. wcel 1767   E*wmo 2276   F/_wnfc 2615   A.wral 2814   E*wrmo 2817   {crab 2818   _Vcvv 3113   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   Rel wrel 5004   Fun wfun 5582   ` cfv 5588
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 5551  df-fun 5590  df-fn 5591  df-fv 5596
This theorem is referenced by:  funcnv5mpt  27211
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