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Theorem bnj1383 29205
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 229 . 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 229 . 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 229 . 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 29204 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321   <.cop 3819   U.cuni 4017   dom cdm 4880    |` cres 4882   Fun wfun 5450
This theorem is referenced by:  bnj1385  29206  bnj60  29433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pr 4405
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 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-res 4892  df-iota 5420  df-fun 5458  df-fv 5464
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