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Theorem funcnvmpt 28137
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 5218 . . . 4  |-  Rel  `' F
2 nfcv 2582 . . . . 5  |-  F/_ y `' F
3 funcnvmpt.2 . . . . . 6  |-  F/_ x F
43nfcnv 5024 . . . . 5  |-  F/_ x `' F
52, 4dffun6f 5606 . . . 4  |-  ( Fun  `' F  <->  ( Rel  `' F  /\  A. y E* x  y `' F x ) )
61, 5mpbiran 926 . . 3  |-  ( Fun  `' F  <->  A. y E* x  y `' F x )
7 vex 3081 . . . . . 6  |-  y  e. 
_V
8 vex 3081 . . . . . 6  |-  x  e. 
_V
97, 8brcnv 5028 . . . . 5  |-  ( y `' F x  <->  x F
y )
109mobii 2287 . . . 4  |-  ( E* x  y `' F x 
<->  E* x  x F y )
1110albii 1687 . . 3  |-  ( A. y E* x  y `' F x  <->  A. y E* x  x F
y )
126, 11bitri 252 . 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 5629 . . . . . . . . 9  |-  Fun  F
16 funbrfv2b 5916 . . . . . . . . 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 3087 . . . . . . . . . . . . . . 15  |-  ( B  e.  V  ->  B  e.  _V )
2018, 19syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  _V )
2120ex 435 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  A  ->  B  e.  _V )
)
2213, 21ralrimi 2823 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  A  B  e.  _V )
23 funcnvmpt.1 . . . . . . . . . . . . 13  |-  F/_ x A
2423rabid2f 27998 . . . . . . . . . . . 12  |-  ( A  =  { x  e.  A  |  B  e. 
_V }  <->  A. x  e.  A  B  e.  _V )
2522, 24sylibr 215 . . . . . . . . . . 11  |-  ( ph  ->  A  =  { x  e.  A  |  B  e.  _V } )
2614dmmpt 5341 . . . . . . . . . . 11  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
2725, 26syl6reqr 2480 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  A )
2827eleq2d 2490 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  dom  F  <-> 
x  e.  A ) )
2928anbi1d 709 . . . . . . . 8  |-  ( ph  ->  ( ( x  e. 
dom  F  /\  ( F `  x )  =  y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
3017, 29syl5bb 260 . . . . . . 7  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  ( F `  x
)  =  y ) ) )
3130bian1d 27961 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  x F y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
32 simpr 462 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
3314fveq1i 5873 . . . . . . . . . . 11  |-  ( F `
 x )  =  ( ( x  e.  A  |->  B ) `  x )
3423fvmpt2f 5956 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3533, 34syl5eq 2473 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( F `  x
)  =  B )
3632, 18, 35syl2anc 665 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
3736eqeq2d 2434 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  y  =  B ) )
38 eqcom 2429 . . . . . . . . 9  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
3928biimpar 487 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  dom  F )
40 funbrfvb 5914 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  y  <-> 
x F y ) )
4115, 39, 40sylancr 667 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  =  y  <->  x F
y ) )
4238, 41syl5bbr 262 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
4337, 42bitr3d 258 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  B  <->  x F
y ) )
4443pm5.32da 645 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  x F y ) ) )
4531, 44, 303bitr4rd 289 . . . . 5  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  y  =  B
) ) )
4613, 45mobid 2283 . . . 4  |-  ( ph  ->  ( E* x  x F y  <->  E* x
( x  e.  A  /\  y  =  B
) ) )
47 df-rmo 2781 . . . 4  |-  ( E* x  e.  A  y  =  B  <->  E* x
( x  e.  A  /\  y  =  B
) )
4846, 47syl6bbr 266 . . 3  |-  ( ph  ->  ( E* x  x F y  <->  E* x  e.  A  y  =  B ) )
4948albidv 1757 . 2  |-  ( ph  ->  ( A. y E* x  x F y  <->  A. y E* x  e.  A  y  =  B ) )
5012, 49syl5bb 260 1  |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x  e.  A  y  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435    = wceq 1437   F/wnf 1663    e. wcel 1867   E*wmo 2264   F/_wnfc 2568   A.wral 2773   E*wrmo 2776   {crab 2777   _Vcvv 3078   class class class wbr 4417    |-> cmpt 4475   `'ccnv 4844   dom cdm 4845   Rel wrel 4850   Fun wfun 5586   ` cfv 5592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  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 5556  df-fun 5594  df-fn 5595  df-fv 5600
This theorem is referenced by:  funcnv5mpt  28138
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