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Theorem bnj1386 31825
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1386.1  |-  ( ph  <->  A. f  e.  A  Fun  f )
bnj1386.2  |-  D  =  ( dom  f  i^i 
dom  g )
bnj1386.3  |-  ( ps  <->  (
ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D )
) )
bnj1386.4  |-  ( x  e.  A  ->  A. f  x  e.  A )
Assertion
Ref Expression
bnj1386  |-  ( ps 
->  Fun  U. A )
Distinct variable groups:    A, g, x    f, g, x
Allowed substitution hints:    ph( x, f, g)    ps( x, f, g)    A( f)    D( x, f, g)

Proof of Theorem bnj1386
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 bnj1386.1 . 2  |-  ( ph  <->  A. f  e.  A  Fun  f )
2 bnj1386.2 . 2  |-  D  =  ( dom  f  i^i 
dom  g )
3 bnj1386.3 . 2  |-  ( ps  <->  (
ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D )
) )
4 bnj1386.4 . 2  |-  ( x  e.  A  ->  A. f  x  e.  A )
5 biid 236 . 2  |-  ( A. h  e.  A  Fun  h 
<-> 
A. h  e.  A  Fun  h )
6 eqid 2442 . 2  |-  ( dom  h  i^i  dom  g
)  =  ( dom  h  i^i  dom  g
)
7 biid 236 . 2  |-  ( ( A. h  e.  A  Fun  h  /\  A. h  e.  A  A. g  e.  A  ( h  |`  ( dom  h  i^i 
dom  g ) )  =  ( g  |`  ( dom  h  i^i  dom  g ) ) )  <-> 
( A. h  e.  A  Fun  h  /\  A. h  e.  A  A. g  e.  A  (
h  |`  ( dom  h  i^i  dom  g ) )  =  ( g  |`  ( dom  h  i^i  dom  g ) ) ) )
81, 2, 3, 4, 5, 6, 7bnj1385 31824 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1367    = wceq 1369    e. wcel 1756   A.wral 2714    i^i cin 3326   U.cuni 4090   dom cdm 4839    |` cres 4841   Fun wfun 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-iota 5380  df-fun 5419  df-fv 5425
This theorem is referenced by:  bnj1384  32021
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