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Theorem bnj1498 28770
Description: Technical lemma for bnj60 28771. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1498.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1498.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1498.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1498.4  |-  F  = 
U. C
Assertion
Ref Expression
bnj1498  |-  ( R 
FrSe  A  ->  dom  F  =  A )
Distinct variable groups:    A, d,
f, x    B, f    G, d, f, x    R, d, f, x
Allowed substitution hints:    B( x, d)    C( x, f, d)    F( x, f, d)    Y( x, f, d)

Proof of Theorem bnj1498
Dummy variables  t 
z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eliun 4041 . . . . . . 7  |-  ( z  e.  U_ f  e.  C  dom  f  <->  E. f  e.  C  z  e.  dom  f )
2 bnj1498.3 . . . . . . . . . . . . . . . 16  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
32bnj1436 28551 . . . . . . . . . . . . . . 15  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
43bnj1299 28530 . . . . . . . . . . . . . 14  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
5 fndm 5486 . . . . . . . . . . . . . 14  |-  ( f  Fn  d  ->  dom  f  =  d )
64, 5bnj31 28424 . . . . . . . . . . . . 13  |-  ( f  e.  C  ->  E. d  e.  B  dom  f  =  d )
76bnj1196 28506 . . . . . . . . . . . 12  |-  ( f  e.  C  ->  E. d
( d  e.  B  /\  dom  f  =  d ) )
8 bnj1498.1 . . . . . . . . . . . . . . 15  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
98bnj1436 28551 . . . . . . . . . . . . . 14  |-  ( d  e.  B  ->  (
d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) )
109simpld 446 . . . . . . . . . . . . 13  |-  ( d  e.  B  ->  d  C_  A )
1110anim1i 552 . . . . . . . . . . . 12  |-  ( ( d  e.  B  /\  dom  f  =  d
)  ->  ( d  C_  A  /\  dom  f  =  d ) )
127, 11bnj593 28453 . . . . . . . . . . 11  |-  ( f  e.  C  ->  E. d
( d  C_  A  /\  dom  f  =  d ) )
13 sseq1 3314 . . . . . . . . . . . 12  |-  ( dom  f  =  d  -> 
( dom  f  C_  A 
<->  d  C_  A )
)
1413biimparc 474 . . . . . . . . . . 11  |-  ( ( d  C_  A  /\  dom  f  =  d
)  ->  dom  f  C_  A )
1512, 14bnj593 28453 . . . . . . . . . 10  |-  ( f  e.  C  ->  E. d dom  f  C_  A )
1615bnj937 28482 . . . . . . . . 9  |-  ( f  e.  C  ->  dom  f  C_  A )
1716sselda 3293 . . . . . . . 8  |-  ( ( f  e.  C  /\  z  e.  dom  f )  ->  z  e.  A
)
1817rexlimiva 2770 . . . . . . 7  |-  ( E. f  e.  C  z  e.  dom  f  -> 
z  e.  A )
191, 18sylbi 188 . . . . . 6  |-  ( z  e.  U_ f  e.  C  dom  f  -> 
z  e.  A )
202bnj1317 28533 . . . . . . 7  |-  ( w  e.  C  ->  A. f  w  e.  C )
2120bnj1400 28547 . . . . . 6  |-  dom  U. C  =  U_ f  e.  C  dom  f
2219, 21eleq2s 2481 . . . . 5  |-  ( z  e.  dom  U. C  ->  z  e.  A )
23 bnj1498.4 . . . . . 6  |-  F  = 
U. C
2423dmeqi 5013 . . . . 5  |-  dom  F  =  dom  U. C
2522, 24eleq2s 2481 . . . 4  |-  ( z  e.  dom  F  -> 
z  e.  A )
2625ssriv 3297 . . 3  |-  dom  F  C_  A
2726a1i 11 . 2  |-  ( R 
FrSe  A  ->  dom  F  C_  A )
28 bnj1498.2 . . . . . . . 8  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
298, 28, 2bnj1493 28768 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
30 vex 2904 . . . . . . . . . . . 12  |-  x  e. 
_V
3130snid 3786 . . . . . . . . . . 11  |-  x  e. 
{ x }
32 elun1 3459 . . . . . . . . . . 11  |-  ( x  e.  { x }  ->  x  e.  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
3331, 32ax-mp 8 . . . . . . . . . 10  |-  x  e.  ( { x }  u.  trCl ( x ,  A ,  R ) )
34 eleq2 2450 . . . . . . . . . 10  |-  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  -> 
( x  e.  dom  f 
<->  x  e.  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
3533, 34mpbiri 225 . . . . . . . . 9  |-  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  ->  x  e.  dom  f )
3635reximi 2758 . . . . . . . 8  |-  ( E. f  e.  C  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) )  ->  E. f  e.  C  x  e.  dom  f )
3736ralimi 2726 . . . . . . 7  |-  ( A. x  e.  A  E. f  e.  C  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) )  ->  A. x  e.  A  E. f  e.  C  x  e.  dom  f )
3829, 37syl 16 . . . . . 6  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  x  e.  dom  f )
39 eliun 4041 . . . . . . 7  |-  ( x  e.  U_ f  e.  C  dom  f  <->  E. f  e.  C  x  e.  dom  f )
4039ralbii 2675 . . . . . 6  |-  ( A. x  e.  A  x  e.  U_ f  e.  C  dom  f  <->  A. x  e.  A  E. f  e.  C  x  e.  dom  f )
4138, 40sylibr 204 . . . . 5  |-  ( R 
FrSe  A  ->  A. x  e.  A  x  e.  U_ f  e.  C  dom  f )
42 nfcv 2525 . . . . . 6  |-  F/_ x A
438bnj1309 28731 . . . . . . . . 9  |-  ( t  e.  B  ->  A. x  t  e.  B )
442, 43bnj1307 28732 . . . . . . . 8  |-  ( t  e.  C  ->  A. x  t  e.  C )
4544nfcii 2516 . . . . . . 7  |-  F/_ x C
46 nfcv 2525 . . . . . . 7  |-  F/_ x dom  f
4745, 46nfiun 4063 . . . . . 6  |-  F/_ x U_ f  e.  C  dom  f
4842, 47dfss3f 3285 . . . . 5  |-  ( A 
C_  U_ f  e.  C  dom  f  <->  A. x  e.  A  x  e.  U_ f  e.  C  dom  f )
4941, 48sylibr 204 . . . 4  |-  ( R 
FrSe  A  ->  A  C_  U_ f  e.  C  dom  f )
5049, 21syl6sseqr 3340 . . 3  |-  ( R 
FrSe  A  ->  A  C_  dom  U. C )
5150, 24syl6sseqr 3340 . 2  |-  ( R 
FrSe  A  ->  A  C_  dom  F )
5227, 51eqssd 3310 1  |-  ( R 
FrSe  A  ->  dom  F  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {cab 2375   A.wral 2651   E.wrex 2652    u. cun 3263    C_ wss 3265   {csn 3759   <.cop 3762   U.cuni 3959   U_ciun 4037   dom cdm 4820    |` cres 4822    Fn wfn 5391   ` cfv 5396    predc-bnj14 28392    FrSe w-bnj15 28396    trClc-bnj18 28398
This theorem is referenced by:  bnj60  28771
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-reg 7495  ax-inf2 7531
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-1o 6662  df-bnj17 28391  df-bnj14 28393  df-bnj13 28395  df-bnj15 28397  df-bnj18 28399  df-bnj19 28401
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