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Theorem dffun6f 5516
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 5513 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. w E. u A. v ( w A v  ->  v  =  u ) ) )
2 nfcv 2610 . . . . . . 7  |-  F/_ y
w
3 dffun6f.2 . . . . . . 7  |-  F/_ y A
4 nfcv 2610 . . . . . . 7  |-  F/_ y
v
52, 3, 4nfbr 4420 . . . . . 6  |-  F/ y  w A v
6 nfv 1674 . . . . . 6  |-  F/ v  w A y
7 breq2 4380 . . . . . 6  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvmo 2303 . . . . 5  |-  ( E* v  w A v  <->  E* y  w A
y )
98albii 1611 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E* y  w A y )
10 mo2v 2266 . . . . 5  |-  ( E* v  w A v  <->  E. u A. v ( w A v  -> 
v  =  u ) )
1110albii 1611 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
12 nfcv 2610 . . . . . . 7  |-  F/_ x w
13 dffun6f.1 . . . . . . 7  |-  F/_ x A
14 nfcv 2610 . . . . . . 7  |-  F/_ x
y
1512, 13, 14nfbr 4420 . . . . . 6  |-  F/ x  w A y
1615nfmo 2279 . . . . 5  |-  F/ x E* y  w A
y
17 nfv 1674 . . . . 5  |-  F/ w E* y  x A
y
18 breq1 4379 . . . . . 6  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
1918mobidv 2283 . . . . 5  |-  ( w  =  x  ->  ( E* y  w A
y  <->  E* y  x A y ) )
2016, 17, 19cbval 1970 . . . 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 694 . 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 369   A.wal 1368   E.wex 1587   E*wmo 2260   F/_wnfc 2596   class class class wbr 4376   Rel wrel 4929   Fun wfun 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rab 2801  df-v 3056  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-br 4377  df-opab 4435  df-id 4720  df-cnv 4932  df-co 4933  df-fun 5504
This theorem is referenced by:  dffun6  5517  funopab  5535  funcnvmptOLD  26106  funcnvmpt  26107
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