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Theorem bnj1386 33993
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 2457 . 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 33992 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1393    = wceq 1395    e. wcel 1819   A.wral 2807    i^i cin 3470   U.cuni 4251   dom cdm 5008    |` cres 5010   Fun wfun 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fv 5602
This theorem is referenced by:  bnj1384  34189
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