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Theorem wfrlem11 25480
Description: Lemma for well-founded recursion. The union of all acceptable functions is a function. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfrlem11.1  |-  R  We  A
wfrlem11.2  |-  R Se  A
wfrlem11.3  |-  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 )  =  ( G `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }
wfrlem11.4  |-  F  = 
U. B
Assertion
Ref Expression
wfrlem11  |-  Fun  F
Distinct variable groups:    A, f, x, y    x, F    f, G, x, y    R, f, x, y
Allowed substitution hints:    B( x, y, f)    F( y, f)

Proof of Theorem wfrlem11
Dummy variables  g  h  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem11.3 . . 3  |-  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 )  =  ( G `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }
2 wfrlem11.4 . . 3  |-  F  = 
U. B
31, 2wfrlem6 25475 . 2  |-  Rel  F
42eleq2i 2468 . . . . . . . . 9  |-  ( <.
x ,  u >.  e.  F  <->  <. x ,  u >.  e.  U. B )
5 eluni 3978 . . . . . . . . 9  |-  ( <.
x ,  u >.  e. 
U. B  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  B ) )
64, 5bitri 241 . . . . . . . 8  |-  ( <.
x ,  u >.  e.  F  <->  E. g ( <.
x ,  u >.  e.  g  /\  g  e.  B ) )
7 df-br 4173 . . . . . . . 8  |-  ( x F u  <->  <. x ,  u >.  e.  F
)
8 df-br 4173 . . . . . . . . . 10  |-  ( x g u  <->  <. x ,  u >.  e.  g
)
98anbi1i 677 . . . . . . . . 9  |-  ( ( x g u  /\  g  e.  B )  <->  (
<. x ,  u >.  e.  g  /\  g  e.  B ) )
109exbii 1589 . . . . . . . 8  |-  ( E. g ( x g u  /\  g  e.  B )  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  B ) )
116, 7, 103bitr4i 269 . . . . . . 7  |-  ( x F u  <->  E. g
( x g u  /\  g  e.  B
) )
122eleq2i 2468 . . . . . . . . 9  |-  ( <.
x ,  v >.  e.  F  <->  <. x ,  v
>.  e.  U. B )
13 eluni 3978 . . . . . . . . 9  |-  ( <.
x ,  v >.  e.  U. B  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  B ) )
1412, 13bitri 241 . . . . . . . 8  |-  ( <.
x ,  v >.  e.  F  <->  E. h ( <.
x ,  v >.  e.  h  /\  h  e.  B ) )
15 df-br 4173 . . . . . . . 8  |-  ( x F v  <->  <. x ,  v >.  e.  F
)
16 df-br 4173 . . . . . . . . . 10  |-  ( x h v  <->  <. x ,  v >.  e.  h
)
1716anbi1i 677 . . . . . . . . 9  |-  ( ( x h v  /\  h  e.  B )  <->  (
<. x ,  v >.  e.  h  /\  h  e.  B ) )
1817exbii 1589 . . . . . . . 8  |-  ( E. h ( x h v  /\  h  e.  B )  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  B ) )
1914, 15, 183bitr4i 269 . . . . . . 7  |-  ( x F v  <->  E. h
( x h v  /\  h  e.  B
) )
2011, 19anbi12i 679 . . . . . 6  |-  ( ( x F u  /\  x F v )  <->  ( E. g ( x g u  /\  g  e.  B )  /\  E. h ( x h v  /\  h  e.  B ) ) )
21 eeanv 1933 . . . . . 6  |-  ( E. g E. h ( ( x g u  /\  g  e.  B
)  /\  ( x h v  /\  h  e.  B ) )  <->  ( E. g ( x g u  /\  g  e.  B )  /\  E. h ( x h v  /\  h  e.  B ) ) )
2220, 21bitr4i 244 . . . . 5  |-  ( ( x F u  /\  x F v )  <->  E. g E. h ( ( x g u  /\  g  e.  B )  /\  (
x h v  /\  h  e.  B )
) )
23 wfrlem11.1 . . . . . . . . 9  |-  R  We  A
24 wfrlem11.2 . . . . . . . . 9  |-  R Se  A
2523, 24, 1wfrlem5 25474 . . . . . . . 8  |-  ( ( g  e.  B  /\  h  e.  B )  ->  ( ( x g u  /\  x h v )  ->  u  =  v ) )
2625impcom 420 . . . . . . 7  |-  ( ( ( x g u  /\  x h v )  /\  ( g  e.  B  /\  h  e.  B ) )  ->  u  =  v )
2726an4s 800 . . . . . 6  |-  ( ( ( x g u  /\  g  e.  B
)  /\  ( x h v  /\  h  e.  B ) )  ->  u  =  v )
2827exlimivv 1642 . . . . 5  |-  ( E. g E. h ( ( x g u  /\  g  e.  B
)  /\  ( x h v  /\  h  e.  B ) )  ->  u  =  v )
2922, 28sylbi 188 . . . 4  |-  ( ( x F u  /\  x F v )  ->  u  =  v )
3029ax-gen 1552 . . 3  |-  A. v
( ( x F u  /\  x F v )  ->  u  =  v )
3130gen2 1553 . 2  |-  A. x A. u A. v ( ( x F u  /\  x F v )  ->  u  =  v )
32 dffun2 5423 . 2  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x A. u A. v ( ( x F u  /\  x F v )  ->  u  =  v )
) )
333, 31, 32mpbir2an 887 1  |-  Fun  F
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666    C_ wss 3280   <.cop 3777   U.cuni 3975   class class class wbr 4172   Se wse 4499    We wwe 4500    |` cres 4839   Rel wrel 4842   Fun wfun 5407    Fn wfn 5408   ` cfv 5413   Predcpred 25381
This theorem is referenced by:  wfrlem12  25481  wfrlem13  25482  wfr1  25486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-pred 25382
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