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Theorem bnj1386 28611
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 227 . 2  |-  ( A. h  e.  A  Fun  h 
<-> 
A. h  e.  A  Fun  h )
6 eqid 2358 . 2  |-  ( dom  h  i^i  dom  g
)  =  ( dom  h  i^i  dom  g
)
7 biid 227 . 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 28610 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1540    = wceq 1642    e. wcel 1710   A.wral 2619    i^i cin 3227   U.cuni 3906   dom cdm 4768    |` cres 4770   Fun wfun 5328
This theorem is referenced by:  bnj1384  28807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-res 4780  df-iota 5298  df-fun 5336  df-fv 5342
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