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Theorem funcnvmptOLD 28322
Description: Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
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
funcnvmptOLD  |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x ( 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 funcnvmptOLD
StepHypRef Expression
1 relcnv 5229 . . . 4  |-  Rel  `' F
2 nfcv 2603 . . . . 5  |-  F/_ y `' F
3 funcnvmpt.2 . . . . . 6  |-  F/_ x F
43nfcnv 5035 . . . . 5  |-  F/_ x `' F
52, 4dffun6f 5619 . . . 4  |-  ( Fun  `' F  <->  ( Rel  `' F  /\  A. y E* x  y `' F x ) )
61, 5mpbiran 934 . . 3  |-  ( Fun  `' F  <->  A. y E* x  y `' F x )
7 vex 3060 . . . . . 6  |-  y  e. 
_V
8 vex 3060 . . . . . 6  |-  x  e. 
_V
97, 8brcnv 5039 . . . . 5  |-  ( y `' F x  <->  x F
y )
109mobii 2333 . . . 4  |-  ( E* x  y `' F x 
<->  E* x  x F y )
1110albii 1702 . . 3  |-  ( A. y E* x  y `' F x  <->  A. y E* x  x F
y )
126, 11bitri 257 . 2  |-  ( Fun  `' F  <->  A. y E* x  x F y )
13 nfv 1772 . . 3  |-  F/ y
ph
14 funcnvmpt.0 . . . 4  |-  F/ x ph
15 funmpt 5641 . . . . . . . . 9  |-  Fun  (
x  e.  A  |->  B )
16 funcnvmpt.3 . . . . . . . . . 10  |-  F  =  ( x  e.  A  |->  B )
1716funeqi 5625 . . . . . . . . 9  |-  ( Fun 
F  <->  Fun  ( x  e.  A  |->  B ) )
1815, 17mpbir 214 . . . . . . . 8  |-  Fun  F
19 funbrfv2b 5936 . . . . . . . 8  |-  ( Fun 
F  ->  ( x F y  <->  ( x  e.  dom  F  /\  ( F `  x )  =  y ) ) )
2018, 19ax-mp 5 . . . . . . 7  |-  ( x F y  <->  ( x  e.  dom  F  /\  ( F `  x )  =  y ) )
21 funcnvmpt.4 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
22 elex 3066 . . . . . . . . . . . . . 14  |-  ( B  e.  V  ->  B  e.  _V )
2321, 22syl 17 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  _V )
2423ex 440 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  A  ->  B  e.  _V )
)
2514, 24ralrimi 2800 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  A  B  e.  _V )
26 funcnvmpt.1 . . . . . . . . . . . 12  |-  F/_ x A
2726rabid2f 28192 . . . . . . . . . . 11  |-  ( A  =  { x  e.  A  |  B  e. 
_V }  <->  A. x  e.  A  B  e.  _V )
2825, 27sylibr 217 . . . . . . . . . 10  |-  ( ph  ->  A  =  { x  e.  A  |  B  e.  _V } )
2916dmmpt 5353 . . . . . . . . . 10  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
3028, 29syl6reqr 2515 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  A )
3130eleq2d 2525 . . . . . . . 8  |-  ( ph  ->  ( x  e.  dom  F  <-> 
x  e.  A ) )
3231anbi1d 716 . . . . . . 7  |-  ( ph  ->  ( ( x  e. 
dom  F  /\  ( F `  x )  =  y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
3320, 32syl5bb 265 . . . . . 6  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  ( F `  x
)  =  y ) ) )
3433bian1d 28156 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  x F y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
3516fveq1i 5893 . . . . . . . . . 10  |-  ( F `
 x )  =  ( ( x  e.  A  |->  B ) `  x )
36 simpr 467 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
3726fvmpt2f 5977 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3836, 21, 37syl2anc 671 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
3935, 38syl5eq 2508 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
4039eqeq2d 2472 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  y  =  B ) )
4131biimpar 492 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  dom  F )
42 funbrfvb 5934 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  y  <-> 
x F y ) )
4318, 41, 42sylancr 674 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  =  y  <->  x F
y ) )
44 eqcom 2469 . . . . . . . . . . 11  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
4544bibi1i 320 . . . . . . . . . 10  |-  ( ( ( F `  x
)  =  y  <->  x F
y )  <->  ( y  =  ( F `  x )  <->  x F
y ) )
4645imbi2i 318 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  ->  (
( F `  x
)  =  y  <->  x F
y ) )  <->  ( ( ph  /\  x  e.  A
)  ->  ( y  =  ( F `  x )  <->  x F
y ) ) )
4743, 46mpbi 213 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
4840, 47bitr3d 263 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  B  <->  x F
y ) )
4948ex 440 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  ( y  =  B  <-> 
x F y ) ) )
5049pm5.32d 649 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  x F y ) ) )
5134, 50, 333bitr4rd 294 . . . 4  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  y  =  B
) ) )
5214, 51mobid 2329 . . 3  |-  ( ph  ->  ( E* x  x F y  <->  E* x
( x  e.  A  /\  y  =  B
) ) )
5313, 52albid 1974 . 2  |-  ( ph  ->  ( A. y E* x  x F y  <->  A. y E* x ( x  e.  A  /\  y  =  B )
) )
5412, 53syl5bb 265 1  |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x ( x  e.  A  /\  y  =  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1453    = wceq 1455   F/wnf 1678    e. wcel 1898   E*wmo 2311   F/_wnfc 2590   A.wral 2749   {crab 2753   _Vcvv 3057   class class class wbr 4418    |-> cmpt 4477   `'ccnv 4855   dom cdm 4856   Rel wrel 4861   Fun wfun 5599   ` cfv 5605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-fv 5613
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator