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Theorem funcnvmptOLD 24074
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 5234 . . . 4  |-  Rel  `' F
2 nfcv 2571 . . . . 5  |-  F/_ y `' F
3 funcnvmpt.2 . . . . . 6  |-  F/_ x F
43nfcnv 5043 . . . . 5  |-  F/_ x `' F
52, 4dffun6f 5460 . . . 4  |-  ( Fun  `' F  <->  ( Rel  `' F  /\  A. y E* x  y `' F x ) )
61, 5mpbiran 885 . . 3  |-  ( Fun  `' F  <->  A. y E* x  y `' F x )
7 vex 2951 . . . . . 6  |-  y  e. 
_V
8 vex 2951 . . . . . 6  |-  x  e. 
_V
97, 8brcnv 5047 . . . . 5  |-  ( y `' F x  <->  x F
y )
109mobii 2316 . . . 4  |-  ( E* x  y `' F x 
<->  E* x  x F y )
1110albii 1575 . . 3  |-  ( A. y E* x  y `' F x  <->  A. y E* x  x F
y )
126, 11bitri 241 . 2  |-  ( Fun  `' F  <->  A. y E* x  x F y )
13 nfv 1629 . . 3  |-  F/ y
ph
14 funcnvmpt.0 . . . 4  |-  F/ x ph
15 funmpt 5481 . . . . . . . . 9  |-  Fun  (
x  e.  A  |->  B )
16 funcnvmpt.3 . . . . . . . . . 10  |-  F  =  ( x  e.  A  |->  B )
1716funeqi 5466 . . . . . . . . 9  |-  ( Fun 
F  <->  Fun  ( x  e.  A  |->  B ) )
1815, 17mpbir 201 . . . . . . . 8  |-  Fun  F
19 funbrfv2b 5763 . . . . . . . 8  |-  ( Fun 
F  ->  ( x F y  <->  ( x  e.  dom  F  /\  ( F `  x )  =  y ) ) )
2018, 19ax-mp 8 . . . . . . 7  |-  ( x F y  <->  ( x  e.  dom  F  /\  ( F `  x )  =  y ) )
21 funcnvmpt.4 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
22 elex 2956 . . . . . . . . . . . . . 14  |-  ( B  e.  V  ->  B  e.  _V )
2321, 22syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  _V )
2423ex 424 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  A  ->  B  e.  _V )
)
2514, 24ralrimi 2779 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  A  B  e.  _V )
26 funcnvmpt.1 . . . . . . . . . . . 12  |-  F/_ x A
2726rabid2f 23959 . . . . . . . . . . 11  |-  ( A  =  { x  e.  A  |  B  e. 
_V }  <->  A. x  e.  A  B  e.  _V )
2825, 27sylibr 204 . . . . . . . . . 10  |-  ( ph  ->  A  =  { x  e.  A  |  B  e.  _V } )
2916dmmpt 5357 . . . . . . . . . 10  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
3028, 29syl6reqr 2486 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  A )
3130eleq2d 2502 . . . . . . . 8  |-  ( ph  ->  ( x  e.  dom  F  <-> 
x  e.  A ) )
3231anbi1d 686 . . . . . . 7  |-  ( ph  ->  ( ( x  e. 
dom  F  /\  ( F `  x )  =  y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
3320, 32syl5bb 249 . . . . . 6  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  ( F `  x
)  =  y ) ) )
3433bian1d 23942 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  x F y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
3516fveq1i 5721 . . . . . . . . . 10  |-  ( F `
 x )  =  ( ( x  e.  A  |->  B ) `  x )
36 simpr 448 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
3726fvmpt2f 24064 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3836, 21, 37syl2anc 643 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
3935, 38syl5eq 2479 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
4039eqeq2d 2446 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  y  =  B ) )
4131biimpar 472 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  dom  F )
42 funbrfvb 5761 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  y  <-> 
x F y ) )
4318, 41, 42sylancr 645 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  =  y  <->  x F
y ) )
44 eqcom 2437 . . . . . . . . . . 11  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
4544bibi1i 306 . . . . . . . . . 10  |-  ( ( ( F `  x
)  =  y  <->  x F
y )  <->  ( y  =  ( F `  x )  <->  x F
y ) )
4645imbi2i 304 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  ->  (
( F `  x
)  =  y  <->  x F
y ) )  <->  ( ( ph  /\  x  e.  A
)  ->  ( y  =  ( F `  x )  <->  x F
y ) ) )
4743, 46mpbi 200 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
4840, 47bitr3d 247 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  B  <->  x F
y ) )
4948ex 424 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  ( y  =  B  <-> 
x F y ) ) )
5049pm5.32d 621 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  x F y ) ) )
5134, 50, 333bitr4rd 278 . . . 4  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  y  =  B
) ) )
5214, 51mobid 2314 . . 3  |-  ( ph  ->  ( E* x  x F y  <->  E* x
( x  e.  A  /\  y  =  B
) ) )
5313, 52albid 1788 . 2  |-  ( ph  ->  ( A. y E* x  x F y  <->  A. y E* x ( x  e.  A  /\  y  =  B )
) )
5412, 53syl5bb 249 1  |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x ( x  e.  A  /\  y  =  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   F/wnf 1553    = wceq 1652    e. wcel 1725   E*wmo 2281   F/_wnfc 2558   A.wral 2697   {crab 2701   _Vcvv 2948   class class class wbr 4204    e. cmpt 4258   `'ccnv 4869   dom cdm 4870   Rel wrel 4875   Fun wfun 5440   ` cfv 5446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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