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Theorem bnj1498 33071
Description: Technical lemma for bnj60 33072. 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 4323 . . . . . . 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 32852 . . . . . . . . . . . . . . 15  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
43bnj1299 32831 . . . . . . . . . . . . . 14  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
5 fndm 5671 . . . . . . . . . . . . . 14  |-  ( f  Fn  d  ->  dom  f  =  d )
64, 5bnj31 32727 . . . . . . . . . . . . 13  |-  ( f  e.  C  ->  E. d  e.  B  dom  f  =  d )
76bnj1196 32807 . . . . . . . . . . . 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 32852 . . . . . . . . . . . . . 14  |-  ( d  e.  B  ->  (
d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) )
109simpld 459 . . . . . . . . . . . . 13  |-  ( d  e.  B  ->  d  C_  A )
1110anim1i 568 . . . . . . . . . . . 12  |-  ( ( d  e.  B  /\  dom  f  =  d
)  ->  ( d  C_  A  /\  dom  f  =  d ) )
127, 11bnj593 32756 . . . . . . . . . . 11  |-  ( f  e.  C  ->  E. d
( d  C_  A  /\  dom  f  =  d ) )
13 sseq1 3518 . . . . . . . . . . . 12  |-  ( dom  f  =  d  -> 
( dom  f  C_  A 
<->  d  C_  A )
)
1413biimparc 487 . . . . . . . . . . 11  |-  ( ( d  C_  A  /\  dom  f  =  d
)  ->  dom  f  C_  A )
1512, 14bnj593 32756 . . . . . . . . . 10  |-  ( f  e.  C  ->  E. d dom  f  C_  A )
1615bnj937 32784 . . . . . . . . 9  |-  ( f  e.  C  ->  dom  f  C_  A )
1716sselda 3497 . . . . . . . 8  |-  ( ( f  e.  C  /\  z  e.  dom  f )  ->  z  e.  A
)
1817rexlimiva 2944 . . . . . . 7  |-  ( E. f  e.  C  z  e.  dom  f  -> 
z  e.  A )
191, 18sylbi 195 . . . . . 6  |-  ( z  e.  U_ f  e.  C  dom  f  -> 
z  e.  A )
202bnj1317 32834 . . . . . . 7  |-  ( w  e.  C  ->  A. f  w  e.  C )
2120bnj1400 32848 . . . . . 6  |-  dom  U. C  =  U_ f  e.  C  dom  f
2219, 21eleq2s 2568 . . . . 5  |-  ( z  e.  dom  U. C  ->  z  e.  A )
23 bnj1498.4 . . . . . 6  |-  F  = 
U. C
2423dmeqi 5195 . . . . 5  |-  dom  F  =  dom  U. C
2522, 24eleq2s 2568 . . . 4  |-  ( z  e.  dom  F  -> 
z  e.  A )
2625ssriv 3501 . . 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 33069 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
30 ssnid 4049 . . . . . . . . . . 11  |-  x  e. 
{ x }
31 elun1 3664 . . . . . . . . . . 11  |-  ( x  e.  { x }  ->  x  e.  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
3230, 31ax-mp 5 . . . . . . . . . 10  |-  x  e.  ( { x }  u.  trCl ( x ,  A ,  R ) )
33 eleq2 2533 . . . . . . . . . 10  |-  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  -> 
( x  e.  dom  f 
<->  x  e.  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
3432, 33mpbiri 233 . . . . . . . . 9  |-  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  ->  x  e.  dom  f )
3534reximi 2925 . . . . . . . 8  |-  ( E. f  e.  C  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) )  ->  E. f  e.  C  x  e.  dom  f )
3635ralimi 2850 . . . . . . 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 )
3729, 36syl 16 . . . . . 6  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  x  e.  dom  f )
38 eliun 4323 . . . . . . 7  |-  ( x  e.  U_ f  e.  C  dom  f  <->  E. f  e.  C  x  e.  dom  f )
3938ralbii 2888 . . . . . 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 )
4037, 39sylibr 212 . . . . 5  |-  ( R 
FrSe  A  ->  A. x  e.  A  x  e.  U_ f  e.  C  dom  f )
41 nfcv 2622 . . . . . 6  |-  F/_ x A
428bnj1309 33032 . . . . . . . . 9  |-  ( t  e.  B  ->  A. x  t  e.  B )
432, 42bnj1307 33033 . . . . . . . 8  |-  ( t  e.  C  ->  A. x  t  e.  C )
4443nfcii 2612 . . . . . . 7  |-  F/_ x C
45 nfcv 2622 . . . . . . 7  |-  F/_ x dom  f
4644, 45nfiun 4346 . . . . . 6  |-  F/_ x U_ f  e.  C  dom  f
4741, 46dfss3f 3489 . . . . 5  |-  ( A 
C_  U_ f  e.  C  dom  f  <->  A. x  e.  A  x  e.  U_ f  e.  C  dom  f )
4840, 47sylibr 212 . . . 4  |-  ( R 
FrSe  A  ->  A  C_  U_ f  e.  C  dom  f )
4948, 21syl6sseqr 3544 . . 3  |-  ( R 
FrSe  A  ->  A  C_  dom  U. C )
5049, 24syl6sseqr 3544 . 2  |-  ( R 
FrSe  A  ->  A  C_  dom  F )
5127, 50eqssd 3514 1  |-  ( R 
FrSe  A  ->  dom  F  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   {cab 2445   A.wral 2807   E.wrex 2808    u. cun 3467    C_ wss 3469   {csn 4020   <.cop 4026   U.cuni 4238   U_ciun 4318   dom cdm 4992    |` cres 4994    Fn wfn 5574   ` cfv 5579    predc-bnj14 32695    FrSe w-bnj15 32699    trClc-bnj18 32701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-reg 8007  ax-inf2 8047
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-1o 7120  df-bnj17 32694  df-bnj14 32696  df-bnj13 32698  df-bnj15 32700  df-bnj18 32702  df-bnj19 32704
This theorem is referenced by:  bnj60  33072
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