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Theorem bnj1383 29428
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 239 . 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 239 . 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 239 . 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 29427 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773    i^i cin 3432   <.cop 3999   U.cuni 4213   dom cdm 4845    |` cres 4847   Fun wfun 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5556  df-fun 5594  df-fv 5600
This theorem is referenced by:  bnj1385  29429  bnj60  29656
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