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Theorem funcnvmptOLD 27487
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 5364 . . . 4  |-  Rel  `' F
2 nfcv 2605 . . . . 5  |-  F/_ y `' F
3 funcnvmpt.2 . . . . . 6  |-  F/_ x F
43nfcnv 5171 . . . . 5  |-  F/_ x `' F
52, 4dffun6f 5592 . . . 4  |-  ( Fun  `' F  <->  ( Rel  `' F  /\  A. y E* x  y `' F x ) )
61, 5mpbiran 918 . . 3  |-  ( Fun  `' F  <->  A. y E* x  y `' F x )
7 vex 3098 . . . . . 6  |-  y  e. 
_V
8 vex 3098 . . . . . 6  |-  x  e. 
_V
97, 8brcnv 5175 . . . . 5  |-  ( y `' F x  <->  x F
y )
109mobii 2293 . . . 4  |-  ( E* x  y `' F x 
<->  E* x  x F y )
1110albii 1627 . . 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 nfv 1694 . . 3  |-  F/ y
ph
14 funcnvmpt.0 . . . 4  |-  F/ x ph
15 funmpt 5614 . . . . . . . . 9  |-  Fun  (
x  e.  A  |->  B )
16 funcnvmpt.3 . . . . . . . . . 10  |-  F  =  ( x  e.  A  |->  B )
1716funeqi 5598 . . . . . . . . 9  |-  ( Fun 
F  <->  Fun  ( x  e.  A  |->  B ) )
1815, 17mpbir 209 . . . . . . . 8  |-  Fun  F
19 funbrfv2b 5902 . . . . . . . 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 3104 . . . . . . . . . . . . . 14  |-  ( B  e.  V  ->  B  e.  _V )
2321, 22syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  _V )
2423ex 434 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  e.  A  ->  B  e.  _V )
)
2514, 24ralrimi 2843 . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  A  B  e.  _V )
26 funcnvmpt.1 . . . . . . . . . . . 12  |-  F/_ x A
2726rabid2f 27378 . . . . . . . . . . 11  |-  ( A  =  { x  e.  A  |  B  e. 
_V }  <->  A. x  e.  A  B  e.  _V )
2825, 27sylibr 212 . . . . . . . . . 10  |-  ( ph  ->  A  =  { x  e.  A  |  B  e.  _V } )
2916dmmpt 5492 . . . . . . . . . 10  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
3028, 29syl6reqr 2503 . . . . . . . . 9  |-  ( ph  ->  dom  F  =  A )
3130eleq2d 2513 . . . . . . . 8  |-  ( ph  ->  ( x  e.  dom  F  <-> 
x  e.  A ) )
3231anbi1d 704 . . . . . . 7  |-  ( ph  ->  ( ( x  e. 
dom  F  /\  ( F `  x )  =  y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
3320, 32syl5bb 257 . . . . . 6  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  ( F `  x
)  =  y ) ) )
3433bian1d 27344 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  x F y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
3516fveq1i 5857 . . . . . . . . . 10  |-  ( F `
 x )  =  ( ( x  e.  A  |->  B ) `  x )
36 simpr 461 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
3726fvmpt2f 27476 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3836, 21, 37syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
( x  e.  A  |->  B ) `  x
)  =  B )
3935, 38syl5eq 2496 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
4039eqeq2d 2457 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  y  =  B ) )
4131biimpar 485 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  dom  F )
42 funbrfvb 5900 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  y  <-> 
x F y ) )
4318, 41, 42sylancr 663 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  =  y  <->  x F
y ) )
44 eqcom 2452 . . . . . . . . . . 11  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
4544bibi1i 314 . . . . . . . . . 10  |-  ( ( ( F `  x
)  =  y  <->  x F
y )  <->  ( y  =  ( F `  x )  <->  x F
y ) )
4645imbi2i 312 . . . . . . . . 9  |-  ( ( ( ph  /\  x  e.  A )  ->  (
( F `  x
)  =  y  <->  x F
y ) )  <->  ( ( ph  /\  x  e.  A
)  ->  ( y  =  ( F `  x )  <->  x F
y ) ) )
4743, 46mpbi 208 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
4840, 47bitr3d 255 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  B  <->  x F
y ) )
4948ex 434 . . . . . 6  |-  ( ph  ->  ( x  e.  A  ->  ( y  =  B  <-> 
x F y ) ) )
5049pm5.32d 639 . . . . 5  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  x F y ) ) )
5134, 50, 333bitr4rd 286 . . . 4  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  y  =  B
) ) )
5214, 51mobid 2289 . . 3  |-  ( ph  ->  ( E* x  x F y  <->  E* x
( x  e.  A  /\  y  =  B
) ) )
5313, 52albid 1871 . 2  |-  ( ph  ->  ( A. y E* x  x F y  <->  A. y E* x ( x  e.  A  /\  y  =  B )
) )
5412, 53syl5bb 257 1  |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x ( x  e.  A  /\  y  =  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1381    = wceq 1383   F/wnf 1603    e. wcel 1804   E*wmo 2269   F/_wnfc 2591   A.wral 2793   {crab 2797   _Vcvv 3095   class class class wbr 4437    |-> cmpt 4495   `'ccnv 4988   dom cdm 4989   Rel wrel 4994   Fun wfun 5572   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586
This theorem is referenced by: (None)
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