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Theorem funcnvmpt 28264
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 5206 . . . 4  |-  Rel  `' F
2 nfcv 2591 . . . . 5  |-  F/_ y `' F
3 funcnvmpt.2 . . . . . 6  |-  F/_ x F
43nfcnv 5012 . . . . 5  |-  F/_ x `' F
52, 4dffun6f 5595 . . . 4  |-  ( Fun  `' F  <->  ( Rel  `' F  /\  A. y E* x  y `' F x ) )
61, 5mpbiran 928 . . 3  |-  ( Fun  `' F  <->  A. y E* x  y `' F x )
7 vex 3047 . . . . . 6  |-  y  e. 
_V
8 vex 3047 . . . . . 6  |-  x  e. 
_V
97, 8brcnv 5016 . . . . 5  |-  ( y `' F x  <->  x F
y )
109mobii 2321 . . . 4  |-  ( E* x  y `' F x 
<->  E* x  x F y )
1110albii 1690 . . 3  |-  ( A. y E* x  y `' F x  <->  A. y E* x  x F
y )
126, 11bitri 253 . 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 5618 . . . . . . . . 9  |-  Fun  F
16 funbrfv2b 5907 . . . . . . . . 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 3053 . . . . . . . . . . . . . . 15  |-  ( B  e.  V  ->  B  e.  _V )
2018, 19syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  _V )
2120ex 436 . . . . . . . . . . . . 13  |-  ( ph  ->  ( x  e.  A  ->  B  e.  _V )
)
2213, 21ralrimi 2787 . . . . . . . . . . . 12  |-  ( ph  ->  A. x  e.  A  B  e.  _V )
23 funcnvmpt.1 . . . . . . . . . . . . 13  |-  F/_ x A
2423rabid2f 28129 . . . . . . . . . . . 12  |-  ( A  =  { x  e.  A  |  B  e. 
_V }  <->  A. x  e.  A  B  e.  _V )
2522, 24sylibr 216 . . . . . . . . . . 11  |-  ( ph  ->  A  =  { x  e.  A  |  B  e.  _V } )
2614dmmpt 5329 . . . . . . . . . . 11  |-  dom  F  =  { x  e.  A  |  B  e.  _V }
2725, 26syl6reqr 2503 . . . . . . . . . 10  |-  ( ph  ->  dom  F  =  A )
2827eleq2d 2513 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  dom  F  <-> 
x  e.  A ) )
2928anbi1d 710 . . . . . . . 8  |-  ( ph  ->  ( ( x  e. 
dom  F  /\  ( F `  x )  =  y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
3017, 29syl5bb 261 . . . . . . 7  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  ( F `  x
)  =  y ) ) )
3130bian1d 28093 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  x F y )  <->  ( x  e.  A  /\  ( F `  x )  =  y ) ) )
32 simpr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
3314fveq1i 5864 . . . . . . . . . . 11  |-  ( F `
 x )  =  ( ( x  e.  A  |->  B ) `  x )
3423fvmpt2f 5947 . . . . . . . . . . 11  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
3533, 34syl5eq 2496 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  B  e.  V )  ->  ( F `  x
)  =  B )
3632, 18, 35syl2anc 666 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
3736eqeq2d 2460 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  y  =  B ) )
38 eqcom 2457 . . . . . . . . 9  |-  ( ( F `  x )  =  y  <->  y  =  ( F `  x ) )
3928biimpar 488 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  dom  F )
40 funbrfvb 5905 . . . . . . . . . 10  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  =  y  <-> 
x F y ) )
4115, 39, 40sylancr 668 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  (
( F `  x
)  =  y  <->  x F
y ) )
4238, 41syl5bbr 263 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  ( F `
 x )  <->  x F
y ) )
4337, 42bitr3d 259 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
y  =  B  <->  x F
y ) )
4443pm5.32da 646 . . . . . 6  |-  ( ph  ->  ( ( x  e.  A  /\  y  =  B )  <->  ( x  e.  A  /\  x F y ) ) )
4531, 44, 303bitr4rd 290 . . . . 5  |-  ( ph  ->  ( x F y  <-> 
( x  e.  A  /\  y  =  B
) ) )
4613, 45mobid 2317 . . . 4  |-  ( ph  ->  ( E* x  x F y  <->  E* x
( x  e.  A  /\  y  =  B
) ) )
47 df-rmo 2744 . . . 4  |-  ( E* x  e.  A  y  =  B  <->  E* x
( x  e.  A  /\  y  =  B
) )
4846, 47syl6bbr 267 . . 3  |-  ( ph  ->  ( E* x  x F y  <->  E* x  e.  A  y  =  B ) )
4948albidv 1766 . 2  |-  ( ph  ->  ( A. y E* x  x F y  <->  A. y E* x  e.  A  y  =  B ) )
5012, 49syl5bb 261 1  |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x  e.  A  y  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1441    = wceq 1443   F/wnf 1666    e. wcel 1886   E*wmo 2299   F/_wnfc 2578   A.wral 2736   E*wrmo 2739   {crab 2740   _Vcvv 3044   class class class wbr 4401    |-> cmpt 4460   `'ccnv 4832   dom cdm 4833   Rel wrel 4838   Fun wfun 5575   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-fv 5589
This theorem is referenced by:  funcnv5mpt  28265
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