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Theorem bnj1386 28915
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 228 . 2  |-  ( A. h  e.  A  Fun  h 
<-> 
A. h  e.  A  Fun  h )
6 eqid 2408 . 2  |-  ( dom  h  i^i  dom  g
)  =  ( dom  h  i^i  dom  g
)
7 biid 228 . 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 28914 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   A.wral 2670    i^i cin 3283   U.cuni 3979   dom cdm 4841    |` cres 4843   Fun wfun 5411
This theorem is referenced by:  bnj1384  29111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-res 4853  df-iota 5381  df-fun 5419  df-fv 5425
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