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Theorem dffun6f 5015
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 5012 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. w A. v A. u ( ( w A v  /\  w A u )  -> 
v  =  u ) ) )
2 nfcv 2228 . . . . . . 7  |-  F/_ y
w
3 dffun6f.2 . . . . . . 7  |-  F/_ y A
4 nfcv 2228 . . . . . . 7  |-  F/_ y
v
52, 3, 4nfbr 3881 . . . . . 6  |-  F/ y  w A v
6 nfv 1466 . . . . . 6  |-  F/ v  w A y
7 breq2 3841 . . . . . 6  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvmo 1988 . . . . 5  |-  ( E* v  w A v  <->  E* y  w A
y )
98albii 1404 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E* y  w A y )
10 breq2 3841 . . . . . 6  |-  ( v  =  u  ->  (
w A v  <->  w A u ) )
1110mo4 2009 . . . . 5  |-  ( E* v  w A v  <->  A. v A. u ( ( w A v  /\  w A u )  ->  v  =  u ) )
1211albii 1404 . . . 4  |-  ( A. w E* v  w A v  <->  A. w A. v A. u ( ( w A v  /\  w A u )  -> 
v  =  u ) )
13 nfcv 2228 . . . . . . 7  |-  F/_ x w
14 dffun6f.1 . . . . . . 7  |-  F/_ x A
15 nfcv 2228 . . . . . . 7  |-  F/_ x
y
1613, 14, 15nfbr 3881 . . . . . 6  |-  F/ x  w A y
1716nfmo 1968 . . . . 5  |-  F/ x E* y  w A
y
18 nfv 1466 . . . . 5  |-  F/ w E* y  x A
y
19 breq1 3840 . . . . . 6  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
2019mobidv 1984 . . . . 5  |-  ( w  =  x  ->  ( E* y  w A
y  <->  E* y  x A y ) )
2117, 18, 20cbval 1684 . . . 4  |-  ( A. w E* y  w A y  <->  A. x E* y  x A y )
229, 12, 213bitr3ri 209 . . 3  |-  ( A. x E* y  x A y  <->  A. w A. v A. u ( ( w A v  /\  w A u )  -> 
v  =  u ) )
2322anbi2i 445 . 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 185 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   E*wmo 1949   F/_wnfc 2215   class class class wbr 3837   Rel wrel 4433   Fun wfun 4996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-cnv 4436  df-co 4437  df-fun 5004
This theorem is referenced by:  dffun6  5016  dffun4f  5018  funopab  5035
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