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Theorem bnj1383 29180
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 227 . 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 227 . 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 227 . 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 29179 1  |-  ( ps 
->  Fun  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164   <.cop 3656   U.cuni 3843   dom cdm 4705    |` cres 4707   Fun wfun 5265
This theorem is referenced by:  bnj1385  29181  bnj60  29408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279
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