ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dffun6f Unicode version

Theorem dffun6f 4943
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 4940 . 2  |-  ( Fun 
A  <->  ( Rel  A  /\  A. w A. v A. u ( ( w A v  /\  w A u )  -> 
v  =  u ) ) )
2 nfcv 2194 . . . . . . 7  |-  F/_ y
w
3 dffun6f.2 . . . . . . 7  |-  F/_ y A
4 nfcv 2194 . . . . . . 7  |-  F/_ y
v
52, 3, 4nfbr 3836 . . . . . 6  |-  F/ y  w A v
6 nfv 1437 . . . . . 6  |-  F/ v  w A y
7 breq2 3796 . . . . . 6  |-  ( v  =  y  ->  (
w A v  <->  w A
y ) )
85, 6, 7cbvmo 1956 . . . . 5  |-  ( E* v  w A v  <->  E* y  w A
y )
98albii 1375 . . . 4  |-  ( A. w E* v  w A v  <->  A. w E* y  w A y )
10 breq2 3796 . . . . . 6  |-  ( v  =  u  ->  (
w A v  <->  w A u ) )
1110mo4 1977 . . . . 5  |-  ( E* v  w A v  <->  A. v A. u ( ( w A v  /\  w A u )  ->  v  =  u ) )
1211albii 1375 . . . 4  |-  ( A. w E* v  w A v  <->  A. w A. v A. u ( ( w A v  /\  w A u )  -> 
v  =  u ) )
13 nfcv 2194 . . . . . . 7  |-  F/_ x w
14 dffun6f.1 . . . . . . 7  |-  F/_ x A
15 nfcv 2194 . . . . . . 7  |-  F/_ x
y
1613, 14, 15nfbr 3836 . . . . . 6  |-  F/ x  w A y
1716nfmo 1936 . . . . 5  |-  F/ x E* y  w A
y
18 nfv 1437 . . . . 5  |-  F/ w E* y  x A
y
19 breq1 3795 . . . . . 6  |-  ( w  =  x  ->  (
w A y  <->  x A
y ) )
2019mobidv 1952 . . . . 5  |-  ( w  =  x  ->  ( E* y  w A
y  <->  E* y  x A y ) )
2117, 18, 20cbval 1653 . . . 4  |-  ( A. w E* y  w A y  <->  A. x E* y  x A y )
229, 12, 213bitr3ri 204 . . 3  |-  ( A. x E* y  x A y  <->  A. w A. v A. u ( ( w A v  /\  w A u )  -> 
v  =  u ) )
2322anbi2i 438 . 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 180 1  |-  ( Fun 
A  <->  ( Rel  A  /\  A. x E* y  x A y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257   E*wmo 1917   F/_wnfc 2181   class class class wbr 3792   Rel wrel 4378   Fun wfun 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-cnv 4381  df-co 4382  df-fun 4932
This theorem is referenced by:  dffun6  4944  dffun4f  4946  funopab  4963
  Copyright terms: Public domain W3C validator