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Theorem bnj1383 31837
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 31836 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727    i^i cin 3339   <.cop 3895   U.cuni 4103   dom cdm 4852    |` cres 4854   Fun wfun 5424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-res 4864  df-iota 5393  df-fun 5432  df-fv 5438
This theorem is referenced by:  bnj1385  31838  bnj60  32065
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