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Theorem dffun6f 5614
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 5611 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. w E. u A. v ( w A v  ->  v  =  u ) ) )
2 nfcv 2602 . . . . . . 7  |-  F/_ y
w
3 dffun6f.2 . . . . . . 7  |-  F/_ y A
4 nfcv 2602 . . . . . . 7  |-  F/_ y
v
52, 3, 4nfbr 4460 . . . . . 6  |-  F/ y  w A v
6 nfv 1771 . . . . . 6  |-  F/ v  w A y
7 breq2 4419 . . . . . 6  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvmo 2345 . . . . 5  |-  ( E* v  w A v  <->  E* y  w A
y )
98albii 1701 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E* y  w A y )
10 mo2v 2316 . . . . 5  |-  ( E* v  w A v  <->  E. u A. v ( w A v  -> 
v  =  u ) )
1110albii 1701 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
12 nfcv 2602 . . . . . . 7  |-  F/_ x w
13 dffun6f.1 . . . . . . 7  |-  F/_ x A
14 nfcv 2602 . . . . . . 7  |-  F/_ x
y
1512, 13, 14nfbr 4460 . . . . . 6  |-  F/ x  w A y
1615nfmo 2326 . . . . 5  |-  F/ x E* y  w A
y
17 nfv 1771 . . . . 5  |-  F/ w E* y  x A
y
18 breq1 4418 . . . . . 6  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
1918mobidv 2330 . . . . 5  |-  ( w  =  x  ->  ( E* y  w A
y  <->  E* y  x A y ) )
2016, 17, 19cbval 2124 . . . 4  |-  ( A. w E* y  w A y  <->  A. x E* y  x A y )
219, 11, 203bitr3ri 284 . . 3  |-  ( A. x E* y  x A y  <->  A. w E. u A. v ( w A v  ->  v  =  u ) )
2221anbi2i 705 . 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 260 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   A.wal 1452   E.wex 1673   E*wmo 2310   F/_wnfc 2589   class class class wbr 4415   Rel wrel 4857   Fun wfun 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-br 4416  df-opab 4475  df-id 4767  df-cnv 4860  df-co 4861  df-fun 5602
This theorem is referenced by:  dffun6  5615  funopab  5633  funcnvmptOLD  28318  funcnvmpt  28319
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