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Theorem dffun6f 5269
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 5266 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. w E. u A. v ( w A v  ->  v  =  u ) ) )
2 nfcv 2419 . . . . . . 7  |-  F/_ y
w
3 dffun6f.2 . . . . . . 7  |-  F/_ y A
4 nfcv 2419 . . . . . . 7  |-  F/_ y
v
52, 3, 4nfbr 4067 . . . . . 6  |-  F/ y  w A v
6 nfv 1605 . . . . . 6  |-  F/ v  w A y
7 breq2 4027 . . . . . 6  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvmo 2180 . . . . 5  |-  ( E* v  w A v  <->  E* y  w A
y )
98albii 1553 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E* y  w A y )
10 nfv 1605 . . . . . 6  |-  F/ u  w A v
1110mo2 2172 . . . . 5  |-  ( E* v  w A v  <->  E. u A. v ( w A v  -> 
v  =  u ) )
1211albii 1553 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
13 nfcv 2419 . . . . . . 7  |-  F/_ x w
14 dffun6f.1 . . . . . . 7  |-  F/_ x A
15 nfcv 2419 . . . . . . 7  |-  F/_ x
y
1613, 14, 15nfbr 4067 . . . . . 6  |-  F/ x  w A y
1716nfmo 2160 . . . . 5  |-  F/ x E* y  w A
y
18 nfv 1605 . . . . 5  |-  F/ w E* y  x A
y
19 breq1 4026 . . . . . 6  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
2019mobidv 2178 . . . . 5  |-  ( w  =  x  ->  ( E* y  w A
y  <->  E* y  x A y ) )
2117, 18, 20cbval 1924 . . . 4  |-  ( A. w E* y  w A y  <->  A. x E* y  x A y )
229, 12, 213bitr3ri 267 . . 3  |-  ( A. x E* y  x A y  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
2322anbi2i 675 . 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 243 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623   E*wmo 2144   F/_wnfc 2406   class class class wbr 4023   Rel wrel 4694   Fun wfun 5249
This theorem is referenced by:  dffun6  5270  funopab  5287  funcnvmptOLD  23234  funcnvmpt  23235
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-cnv 4697  df-co 4698  df-fun 5257
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