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Theorem bnj1386 29279
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 229 . 2  |-  ( A. h  e.  A  Fun  h 
<-> 
A. h  e.  A  Fun  h )
6 eqid 2438 . 2  |-  ( dom  h  i^i  dom  g
)  =  ( dom  h  i^i  dom  g
)
7 biid 229 . 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 29278 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321   U.cuni 4017   dom cdm 4881    |` cres 4883   Fun wfun 5451
This theorem is referenced by:  bnj1384  29475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fv 5465
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