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Theorem dffun6f 5468
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 5465 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. w E. u A. v ( w A v  ->  v  =  u ) ) )
2 nfcv 2572 . . . . . . 7  |-  F/_ y
w
3 dffun6f.2 . . . . . . 7  |-  F/_ y A
4 nfcv 2572 . . . . . . 7  |-  F/_ y
v
52, 3, 4nfbr 4256 . . . . . 6  |-  F/ y  w A v
6 nfv 1629 . . . . . 6  |-  F/ v  w A y
7 breq2 4216 . . . . . 6  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvmo 2318 . . . . 5  |-  ( E* v  w A v  <->  E* y  w A
y )
98albii 1575 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E* y  w A y )
10 nfv 1629 . . . . . 6  |-  F/ u  w A v
1110mo2 2310 . . . . 5  |-  ( E* v  w A v  <->  E. u A. v ( w A v  -> 
v  =  u ) )
1211albii 1575 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
13 nfcv 2572 . . . . . . 7  |-  F/_ x w
14 dffun6f.1 . . . . . . 7  |-  F/_ x A
15 nfcv 2572 . . . . . . 7  |-  F/_ x
y
1613, 14, 15nfbr 4256 . . . . . 6  |-  F/ x  w A y
1716nfmo 2298 . . . . 5  |-  F/ x E* y  w A
y
18 nfv 1629 . . . . 5  |-  F/ w E* y  x A
y
19 breq1 4215 . . . . . 6  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
2019mobidv 2316 . . . . 5  |-  ( w  =  x  ->  ( E* y  w A
y  <->  E* y  x A y ) )
2117, 18, 20cbval 1982 . . . 4  |-  ( A. w E* y  w A y  <->  A. x E* y  x A y )
229, 12, 213bitr3ri 268 . . 3  |-  ( A. x E* y  x A y  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
2322anbi2i 676 . 2  |-  ( ( Rel  A  /\  A. x E* y  x A y )  <->  ( Rel  A  /\  A. w E. u A. v ( w A v  ->  v  =  u ) ) )
241, 23bitr4i 244 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550   E*wmo 2282   F/_wnfc 2559   class class class wbr 4212   Rel wrel 4883   Fun wfun 5448
This theorem is referenced by:  dffun6  5469  funopab  5486  funcnvmptOLD  24082  funcnvmpt  24083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-id 4498  df-cnv 4886  df-co 4887  df-fun 5456
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