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Theorem dffun6f 5558
Description: Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dffun6f.1  |-  F/_ x A
dffun6f.2  |-  F/_ y A
Assertion
Ref Expression
dffun6f  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)

Proof of Theorem dffun6f
Dummy variables  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun3 5555 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. w E. u A. v ( w A v  ->  v  =  u ) ) )
2 nfcv 2569 . . . . . . 7  |-  F/_ y
w
3 dffun6f.2 . . . . . . 7  |-  F/_ y A
4 nfcv 2569 . . . . . . 7  |-  F/_ y
v
52, 3, 4nfbr 4411 . . . . . 6  |-  F/ y  w A v
6 nfv 1755 . . . . . 6  |-  F/ v  w A y
7 breq2 4370 . . . . . 6  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvmo 2312 . . . . 5  |-  ( E* v  w A v  <->  E* y  w A
y )
98albii 1685 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E* y  w A y )
10 mo2v 2283 . . . . 5  |-  ( E* v  w A v  <->  E. u A. v ( w A v  -> 
v  =  u ) )
1110albii 1685 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
12 nfcv 2569 . . . . . . 7  |-  F/_ x w
13 dffun6f.1 . . . . . . 7  |-  F/_ x A
14 nfcv 2569 . . . . . . 7  |-  F/_ x
y
1512, 13, 14nfbr 4411 . . . . . 6  |-  F/ x  w A y
1615nfmo 2293 . . . . 5  |-  F/ x E* y  w A
y
17 nfv 1755 . . . . 5  |-  F/ w E* y  x A
y
18 breq1 4369 . . . . . 6  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
1918mobidv 2297 . . . . 5  |-  ( w  =  x  ->  ( E* y  w A
y  <->  E* y  x A y ) )
2016, 17, 19cbval 2086 . . . 4  |-  ( A. w E* y  w A y  <->  A. x E* y  x A y )
219, 11, 203bitr3ri 279 . . 3  |-  ( A. x E* y  x A y  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
2221anbi2i 698 . 2  |-  ( ( Rel  A  /\  A. x E* y  x A y )  <->  ( Rel  A  /\  A. w E. u A. v ( w A v  ->  v  =  u ) ) )
231, 22bitr4i 255 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   A.wal 1435   E.wex 1657   E*wmo 2277   F/_wnfc 2556   class class class wbr 4366   Rel wrel 4801   Fun wfun 5538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-id 4711  df-cnv 4804  df-co 4805  df-fun 5546
This theorem is referenced by:  dffun6  5559  funopab  5577  funcnvmptOLD  28216  funcnvmpt  28217
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