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Theorem dffun6f 5584
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 5581 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. w E. u A. v ( w A v  ->  v  =  u ) ) )
2 nfcv 2616 . . . . . . 7  |-  F/_ y
w
3 dffun6f.2 . . . . . . 7  |-  F/_ y A
4 nfcv 2616 . . . . . . 7  |-  F/_ y
v
52, 3, 4nfbr 4483 . . . . . 6  |-  F/ y  w A v
6 nfv 1712 . . . . . 6  |-  F/ v  w A y
7 breq2 4443 . . . . . 6  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvmo 2323 . . . . 5  |-  ( E* v  w A v  <->  E* y  w A
y )
98albii 1645 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E* y  w A y )
10 mo2v 2291 . . . . 5  |-  ( E* v  w A v  <->  E. u A. v ( w A v  -> 
v  =  u ) )
1110albii 1645 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
12 nfcv 2616 . . . . . . 7  |-  F/_ x w
13 dffun6f.1 . . . . . . 7  |-  F/_ x A
14 nfcv 2616 . . . . . . 7  |-  F/_ x
y
1512, 13, 14nfbr 4483 . . . . . 6  |-  F/ x  w A y
1615nfmo 2303 . . . . 5  |-  F/ x E* y  w A
y
17 nfv 1712 . . . . 5  |-  F/ w E* y  x A
y
18 breq1 4442 . . . . . 6  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
1918mobidv 2307 . . . . 5  |-  ( w  =  x  ->  ( E* y  w A
y  <->  E* y  x A y ) )
2016, 17, 19cbval 2026 . . . 4  |-  ( A. w E* y  w A y  <->  A. x E* y  x A y )
219, 11, 203bitr3ri 276 . . 3  |-  ( A. x E* y  x A y  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
2221anbi2i 692 . 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 252 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367   A.wal 1396   E.wex 1617   E*wmo 2285   F/_wnfc 2602   class class class wbr 4439   Rel wrel 4993   Fun wfun 5564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-cnv 4996  df-co 4997  df-fun 5572
This theorem is referenced by:  dffun6  5585  funopab  5603  funcnvmptOLD  27739  funcnvmpt  27740
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