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Theorem isarep1 5051
Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by  ph ( x ,  y ) i.e. the class  ( {
<. x ,  y >.  |  ph } " A
). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
isarep1  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. x  e.  A  [ b  /  y ] ph )
Distinct variable groups:    x, A    x, b, y
Allowed substitution hints:    ph( x, y, b)    A( y, b)

Proof of Theorem isarep1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 vex 2615 . . 3  |-  b  e. 
_V
21elima 4732 . 2  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. z  e.  A  z { <. x ,  y >.  |  ph } b )
3 df-br 3812 . . . 4  |-  ( z { <. x ,  y
>.  |  ph } b  <->  <. z ,  b >.  e.  { <. x ,  y
>.  |  ph } )
4 opelopabsb 4050 . . . 4  |-  ( <.
z ,  b >.  e.  { <. x ,  y
>.  |  ph }  <->  [. z  /  x ]. [. b  / 
y ]. ph )
5 sbsbc 2830 . . . . . 6  |-  ( [ b  /  y ]
ph 
<-> 
[. b  /  y ]. ph )
65sbbii 1690 . . . . 5  |-  ( [ z  /  x ] [ b  /  y ] ph  <->  [ z  /  x ] [. b  /  y ]. ph )
7 sbsbc 2830 . . . . 5  |-  ( [ z  /  x ] [. b  /  y ]. ph  <->  [. z  /  x ]. [. b  /  y ]. ph )
86, 7bitr2i 183 . . . 4  |-  ( [. z  /  x ]. [. b  /  y ]. ph  <->  [ z  /  x ] [ b  /  y ] ph )
93, 4, 83bitri 204 . . 3  |-  ( z { <. x ,  y
>.  |  ph } b  <->  [ z  /  x ] [ b  /  y ] ph )
109rexbii 2379 . 2  |-  ( E. z  e.  A  z { <. x ,  y
>.  |  ph } b  <->  E. z  e.  A  [ z  /  x ] [ b  /  y ] ph )
11 nfs1v 1858 . . 3  |-  F/ x [ z  /  x ] [ b  /  y ] ph
12 nfv 1462 . . 3  |-  F/ z [ b  /  y ] ph
13 sbequ12r 1697 . . 3  |-  ( z  =  x  ->  ( [ z  /  x ] [ b  /  y ] ph  <->  [ b  /  y ] ph ) )
1411, 12, 13cbvrex 2580 . 2  |-  ( E. z  e.  A  [
z  /  x ] [ b  /  y ] ph  <->  E. x  e.  A  [ b  /  y ] ph )
152, 10, 143bitri 204 1  |-  ( b  e.  ( { <. x ,  y >.  |  ph } " A )  <->  E. x  e.  A  [ b  /  y ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434   [wsb 1687   E.wrex 2354   [.wsbc 2826   <.cop 3425   class class class wbr 3811   {copab 3864   "cima 4402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-xp 4405  df-cnv 4407  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412
This theorem is referenced by: (None)
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