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Theorem tfrlem7 6285
Description: Lemma for transfinite recursion. The union of all acceptable functions is a function. (Contributed by NM, 9-Aug-1994.)
Hypothesis
Ref Expression
tfrlem.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
tfrlem7  |-  Fun recs ( F )
Distinct variable group:    x, f, y, F
Allowed substitution hints:    A( x, y, f)

Proof of Theorem tfrlem7
StepHypRef Expression
1 tfrlem.1 . . 3  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
21tfrlem6 6284 . 2  |-  Rel recs ( F )
31recsfval 6283 . . . . . . . . 9  |- recs ( F )  =  U. A
43eleq2i 2317 . . . . . . . 8  |-  ( <.
x ,  u >.  e. recs
( F )  <->  <. x ,  u >.  e.  U. A
)
5 eluni 3730 . . . . . . . 8  |-  ( <.
x ,  u >.  e. 
U. A  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  A ) )
64, 5bitri 242 . . . . . . 7  |-  ( <.
x ,  u >.  e. recs
( F )  <->  E. g
( <. x ,  u >.  e.  g  /\  g  e.  A ) )
73eleq2i 2317 . . . . . . . 8  |-  ( <.
x ,  v >.  e. recs ( F )  <->  <. x ,  v >.  e.  U. A
)
8 eluni 3730 . . . . . . . 8  |-  ( <.
x ,  v >.  e.  U. A  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) )
97, 8bitri 242 . . . . . . 7  |-  ( <.
x ,  v >.  e. recs ( F )  <->  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) )
106, 9anbi12i 681 . . . . . 6  |-  ( (
<. x ,  u >.  e. recs
( F )  /\  <.
x ,  v >.  e. recs ( F ) )  <-> 
( E. g (
<. x ,  u >.  e.  g  /\  g  e.  A )  /\  E. h ( <. x ,  v >.  e.  h  /\  h  e.  A
) ) )
11 eeanv 2055 . . . . . 6  |-  ( E. g E. h ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  <->  ( E. g ( <. x ,  u >.  e.  g  /\  g  e.  A
)  /\  E. h
( <. x ,  v
>.  e.  h  /\  h  e.  A ) ) )
1210, 11bitr4i 245 . . . . 5  |-  ( (
<. x ,  u >.  e. recs
( F )  /\  <.
x ,  v >.  e. recs ( F ) )  <->  E. g E. h ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) ) )
13 an4 800 . . . . . . . 8  |-  ( ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  <->  ( ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
)  /\  ( g  e.  A  /\  h  e.  A ) ) )
14 ancom 439 . . . . . . . 8  |-  ( ( ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
)  /\  ( g  e.  A  /\  h  e.  A ) )  <->  ( (
g  e.  A  /\  h  e.  A )  /\  ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
) ) )
1513, 14bitri 242 . . . . . . 7  |-  ( ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  <->  ( (
g  e.  A  /\  h  e.  A )  /\  ( <. x ,  u >.  e.  g  /\  <. x ,  v >.  e.  h
) ) )
161tfrlem5 6282 . . . . . . . 8  |-  ( ( g  e.  A  /\  h  e.  A )  ->  ( ( <. x ,  u >.  e.  g  /\  <. x ,  v
>.  e.  h )  ->  u  =  v )
)
1716imp 420 . . . . . . 7  |-  ( ( ( g  e.  A  /\  h  e.  A
)  /\  ( <. x ,  u >.  e.  g  /\  <. x ,  v
>.  e.  h ) )  ->  u  =  v )
1815, 17sylbi 189 . . . . . 6  |-  ( ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  ->  u  =  v )
1918exlimivv 2025 . . . . 5  |-  ( E. g E. h ( ( <. x ,  u >.  e.  g  /\  g  e.  A )  /\  ( <. x ,  v >.  e.  h  /\  h  e.  A ) )  ->  u  =  v )
2012, 19sylbi 189 . . . 4  |-  ( (
<. x ,  u >.  e. recs
( F )  /\  <.
x ,  v >.  e. recs ( F ) )  ->  u  =  v )
2120ax-gen 1536 . . 3  |-  A. v
( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v >.  e. recs ( F ) )  ->  u  =  v )
2221gen2 1541 . 2  |-  A. x A. u A. v ( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v
>.  e. recs ( F ) )  ->  u  =  v )
23 dffun4 5125 . 2  |-  ( Fun recs
( F )  <->  ( Rel recs ( F )  /\  A. x A. u A. v
( ( <. x ,  u >.  e. recs ( F )  /\  <. x ,  v >.  e. recs ( F ) )  ->  u  =  v )
) )
242, 22, 23mpbir2an 891 1  |-  Fun recs ( F )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2239   A.wral 2509   E.wrex 2510   <.cop 3547   U.cuni 3727   Oncon0 4285    |` cres 4582   Rel wrel 4585   Fun wfun 4586    Fn wfn 4587   ` cfv 4592  recscrecs 6273
This theorem is referenced by:  tfrlem9  6287  tfrlem9a  6288  tfrlem10  6289  tfrlem14  6293  tfrlem16  6295  tfr1a  6296  tfr1  6299
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608  df-recs 6274
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