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Theorem frrlem3 28966
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 28964 . . 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 2594 . 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 5678 . . . 4  |-  ( g  Fn  z  ->  dom  g  =  z )
5 simp1 996 . . . 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 3525 . . . . 5  |-  ( dom  g  =  z  -> 
( dom  g  C_  A 
<->  z  C_  A )
)
76biimpar 485 . . . 4  |-  ( ( dom  g  =  z  /\  z  C_  A
)  ->  dom  g  C_  A )
84, 5, 7syl2an 477 . . 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 1698 . 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 195 1  |-  ( g  e.  B  ->  dom  g  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   A.wral 2814    C_ wss 3476   dom cdm 4999    |` cres 5001    Fn wfn 5581   ` cfv 5586  (class class class)co 6282   Predcpred 28820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-ov 6285  df-pred 28821
This theorem is referenced by:  frrlem5  28968  frrlem5d  28971  frrlem7  28974
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