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Theorem frrlem3 25589
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 25587 . . 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 2545 . 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 5547 . . . 4  |-  ( g  Fn  z  ->  dom  g  =  z )
5 simp1 958 . . . 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 3371 . . . . 5  |-  ( dom  g  =  z  -> 
( dom  g  C_  A 
<->  z  C_  A )
)
76biimpar 473 . . . 4  |-  ( ( dom  g  =  z  /\  z  C_  A
)  ->  dom  g  C_  A )
84, 5, 7syl2an 465 . . 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 1645 . 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 189 1  |-  ( g  e.  B  ->  dom  g  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707    C_ wss 3322   dom cdm 4881    |` cres 4883    Fn wfn 5452   ` cfv 5457  (class class class)co 6084   Predcpred 25443
This theorem is referenced by:  frrlem5  25591  frrlem5d  25594  frrlem7  25597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-ov 6087  df-pred 25444
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