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Theorem wfr3 25489
Description: The principle of Well-Founded Recursion, part 3 of 3. Finally, we show that  F is unique. We do this by showing that any function  H with the same properties we proved of  F in wfr1 25486 and wfr2 25487 is identical to  F. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfr3.1  |-  R  We  A
wfr3.2  |-  R Se  A
wfr3.3  |-  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 )
) ) ) }
wfr3.4  |-  F  = 
U. B
Assertion
Ref Expression
wfr3  |-  ( ( H  Fn  A  /\  A. z  e.  A  ( H `  z )  =  ( G `  ( H  |`  Pred ( R ,  A , 
z ) ) ) )  ->  F  =  H )
Distinct variable groups:    A, f, x, y, z    f, F, x, y, z    f, G, x, y, z    x, H, z    R, f, x, y, z
Allowed substitution hints:    B( x, y, z, f)    H( y, f)

Proof of Theorem wfr3
StepHypRef Expression
1 wfr3.1 . . 3  |-  R  We  A
2 wfr3.2 . . 3  |-  R Se  A
31, 2pm3.2i 442 . 2  |-  ( R  We  A  /\  R Se  A )
4 wfr3.3 . . . 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 )  =  ( G `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }
5 wfr3.4 . . . 4  |-  F  = 
U. B
61, 2, 4, 5wfr1 25486 . . 3  |-  F  Fn  A
71, 2, 4, 5wfr2 25487 . . . 4  |-  ( z  e.  A  ->  ( F `  z )  =  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) ) )
87rgen 2731 . . 3  |-  A. z  e.  A  ( F `  z )  =  ( G `  ( F  |`  Pred ( R ,  A ,  z )
) )
96, 8pm3.2i 442 . 2  |-  ( F  Fn  A  /\  A. z  e.  A  ( F `  z )  =  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) ) )
10 wfr3g 25469 . 2  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( F  Fn  A  /\  A. z  e.  A  ( F `  z )  =  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) ) )  /\  ( H  Fn  A  /\  A. z  e.  A  ( H `  z )  =  ( G `  ( H  |`  Pred ( R ,  A , 
z ) ) ) ) )  ->  F  =  H )
113, 9, 10mp3an12 1269 1  |-  ( ( H  Fn  A  /\  A. z  e.  A  ( H `  z )  =  ( G `  ( H  |`  Pred ( R ,  A , 
z ) ) ) )  ->  F  =  H )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649   {cab 2390   A.wral 2666    C_ wss 3280   U.cuni 3975   Se wse 4499    We wwe 4500    |` cres 4839    Fn wfn 5408   ` cfv 5413   Predcpred 25381
This theorem is referenced by:  tfr3ALT  25493
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-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-pred 25382
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