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Theorem frrlem3 24841
Description: Lemma for founded recursion. An acceptable function's domain is a subset of  A. (Contributed by Paul Chapman, 21-Apr-2012.)
Hypothesis
Ref Expression
frrlem1.1  |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x  /\  A. y  e.  x  ( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }
Assertion
Ref Expression
frrlem3  |-  ( g  e.  B  ->  dom  g  C_  A )
Distinct variable groups:    A, f,
g, x, y    f, G, g, x, y    R, f, g, x, y
Allowed substitution hints:    B( x, y, f, g)

Proof of Theorem frrlem3
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frrlem1.1 . . . 4  |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x  /\  A. y  e.  x  ( f `  y
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }
21frrlem1 24839 . . 3  |-  B  =  { g  |  E. z ( g  Fn  z  /\  ( z 
C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w
)  =  ( w G ( g  |`  Pred ( R ,  A ,  w ) ) ) ) ) }
32abeq2i 2465 . 2  |-  ( g  e.  B  <->  E. z
( g  Fn  z  /\  ( z  C_  A  /\  A. w  e.  z 
Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w )  =  ( w G ( g  |`  Pred ( R ,  A ,  w )
) ) ) ) )
4 fndm 5425 . . . 4  |-  ( g  Fn  z  ->  dom  g  =  z )
5 simp1 955 . . . 4  |-  ( ( z  C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w
)  =  ( w G ( g  |`  Pred ( R ,  A ,  w ) ) ) )  ->  z  C_  A )
6 sseq1 3275 . . . . 5  |-  ( dom  g  =  z  -> 
( dom  g  C_  A 
<->  z  C_  A )
)
76biimpar 471 . . . 4  |-  ( ( dom  g  =  z  /\  z  C_  A
)  ->  dom  g  C_  A )
84, 5, 7syl2an 463 . . 3  |-  ( ( g  Fn  z  /\  ( z  C_  A  /\  A. w  e.  z 
Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w )  =  ( w G ( g  |`  Pred ( R ,  A ,  w )
) ) ) )  ->  dom  g  C_  A )
98exlimiv 1634 . 2  |-  ( E. z ( g  Fn  z  /\  ( z 
C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  /\  A. w  e.  z  ( g `  w
)  =  ( w G ( g  |`  Pred ( R ,  A ,  w ) ) ) ) )  ->  dom  g  C_  A )
103, 9sylbi 187 1  |-  ( g  e.  B  ->  dom  g  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1541    = wceq 1642    e. wcel 1710   {cab 2344   A.wral 2619    C_ wss 3228   dom cdm 4771    |` cres 4773    Fn wfn 5332   ` cfv 5337  (class class class)co 5945   Predcpred 24725
This theorem is referenced by:  frrlem5  24843  frrlem5d  24846  frrlem7  24849
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345  df-ov 5948  df-pred 24726
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