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Theorem bnj1383 33186
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1383.1  |-  ( ph  <->  A. f  e.  A  Fun  f )
bnj1383.2  |-  D  =  ( dom  f  i^i 
dom  g )
bnj1383.3  |-  ( ps  <->  (
ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D )
) )
Assertion
Ref Expression
bnj1383  |-  ( ps 
->  Fun  U. A )
Distinct variable groups:    A, f,
g    ph, g
Allowed substitution hints:    ph( f)    ps( f, g)    D( f, g)

Proof of Theorem bnj1383
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1383.1 . 2  |-  ( ph  <->  A. f  e.  A  Fun  f )
2 bnj1383.2 . 2  |-  D  =  ( dom  f  i^i 
dom  g )
3 bnj1383.3 . 2  |-  ( ps  <->  (
ph  /\  A. f  e.  A  A. g  e.  A  ( f  |`  D )  =  ( g  |`  D )
) )
4 biid 236 . 2  |-  ( ( ps  /\  <. x ,  y >.  e.  U. A  /\  <. x ,  z
>.  e.  U. A )  <-> 
( ps  /\  <. x ,  y >.  e.  U. A  /\  <. x ,  z
>.  e.  U. A ) )
5 biid 236 . 2  |-  ( ( ( ps  /\  <. x ,  y >.  e.  U. A  /\  <. x ,  z
>.  e.  U. A )  /\  f  e.  A  /\  <. x ,  y
>.  e.  f )  <->  ( ( ps  /\  <. x ,  y
>.  e.  U. A  /\  <.
x ,  z >.  e.  U. A )  /\  f  e.  A  /\  <.
x ,  y >.  e.  f ) )
6 biid 236 . 2  |-  ( ( ( ( ps  /\  <.
x ,  y >.  e.  U. A  /\  <. x ,  z >.  e.  U. A )  /\  f  e.  A  /\  <. x ,  y >.  e.  f )  /\  g  e.  A  /\  <. x ,  z >.  e.  g )  <->  ( ( ( ps  /\  <. x ,  y >.  e.  U. A  /\  <. x ,  z
>.  e.  U. A )  /\  f  e.  A  /\  <. x ,  y
>.  e.  f )  /\  g  e.  A  /\  <.
x ,  z >.  e.  g ) )
71, 2, 3, 4, 5, 6bnj1379 33185 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814    i^i cin 3475   <.cop 4033   U.cuni 4245   dom cdm 4999    |` cres 5001   Fun wfun 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fv 5596
This theorem is referenced by:  bnj1385  33187  bnj60  33414
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