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Theorem dffun6f 5104
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 dffun2 5101 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. w A. v A. u ( ( w A v  /\  w A u )  -> 
v  =  u ) ) )
2 nfcv 2256 . . . . . . 7  |-  F/_ y
w
3 dffun6f.2 . . . . . . 7  |-  F/_ y A
4 nfcv 2256 . . . . . . 7  |-  F/_ y
v
52, 3, 4nfbr 3942 . . . . . 6  |-  F/ y  w A v
6 nfv 1491 . . . . . 6  |-  F/ v  w A y
7 breq2 3901 . . . . . 6  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvmo 2015 . . . . 5  |-  ( E* v  w A v  <->  E* y  w A
y )
98albii 1429 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E* y  w A y )
10 breq2 3901 . . . . . 6  |-  ( v  =  u  ->  (
w A v  <->  w A u ) )
1110mo4 2036 . . . . 5  |-  ( E* v  w A v  <->  A. v A. u ( ( w A v  /\  w A u )  ->  v  =  u ) )
1211albii 1429 . . . 4  |-  ( A. w E* v  w A v  <->  A. w A. v A. u ( ( w A v  /\  w A u )  -> 
v  =  u ) )
13 nfcv 2256 . . . . . . 7  |-  F/_ x w
14 dffun6f.1 . . . . . . 7  |-  F/_ x A
15 nfcv 2256 . . . . . . 7  |-  F/_ x
y
1613, 14, 15nfbr 3942 . . . . . 6  |-  F/ x  w A y
1716nfmo 1995 . . . . 5  |-  F/ x E* y  w A
y
18 nfv 1491 . . . . 5  |-  F/ w E* y  x A
y
19 breq1 3900 . . . . . 6  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
2019mobidv 2011 . . . . 5  |-  ( w  =  x  ->  ( E* y  w A
y  <->  E* y  x A y ) )
2117, 18, 20cbval 1710 . . . 4  |-  ( A. w E* y  w A y  <->  A. x E* y  x A y )
229, 12, 213bitr3ri 210 . . 3  |-  ( A. x E* y  x A y  <->  A. w A. v A. u ( ( w A v  /\  w A u )  -> 
v  =  u ) )
2322anbi2i 450 . 2  |-  ( ( Rel  A  /\  A. x E* y  x A y )  <->  ( Rel  A  /\  A. w A. v A. u ( ( w A v  /\  w A u )  -> 
v  =  u ) ) )
241, 23bitr4i 186 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312   E*wmo 1976   F/_wnfc 2243   class class class wbr 3897   Rel wrel 4512   Fun wfun 5085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-cnv 4515  df-co 4516  df-fun 5093
This theorem is referenced by:  dffun6  5105  dffun4f  5107  funopab  5126
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