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Theorem bnj1498 33824
Description: Technical lemma for bnj60 33825. 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 4316 . . . . . . 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 33605 . . . . . . . . . . . . . . 15  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
43bnj1299 33584 . . . . . . . . . . . . . 14  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
5 fndm 5666 . . . . . . . . . . . . . 14  |-  ( f  Fn  d  ->  dom  f  =  d )
64, 5bnj31 33480 . . . . . . . . . . . . 13  |-  ( f  e.  C  ->  E. d  e.  B  dom  f  =  d )
76bnj1196 33560 . . . . . . . . . . . 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 33605 . . . . . . . . . . . . . 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 33509 . . . . . . . . . . 11  |-  ( f  e.  C  ->  E. d
( d  C_  A  /\  dom  f  =  d ) )
13 sseq1 3507 . . . . . . . . . . . 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 33509 . . . . . . . . . 10  |-  ( f  e.  C  ->  E. d dom  f  C_  A )
1615bnj937 33537 . . . . . . . . 9  |-  ( f  e.  C  ->  dom  f  C_  A )
1716sselda 3486 . . . . . . . 8  |-  ( ( f  e.  C  /\  z  e.  dom  f )  ->  z  e.  A
)
1817rexlimiva 2929 . . . . . . 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 33587 . . . . . . 7  |-  ( w  e.  C  ->  A. f  w  e.  C )
2120bnj1400 33601 . . . . . 6  |-  dom  U. C  =  U_ f  e.  C  dom  f
2219, 21eleq2s 2549 . . . . 5  |-  ( z  e.  dom  U. C  ->  z  e.  A )
23 bnj1498.4 . . . . . 6  |-  F  = 
U. C
2423dmeqi 5190 . . . . 5  |-  dom  F  =  dom  U. C
2522, 24eleq2s 2549 . . . 4  |-  ( z  e.  dom  F  -> 
z  e.  A )
2625ssriv 3490 . . 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 33822 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
30 ssnid 4039 . . . . . . . . . . 11  |-  x  e. 
{ x }
31 elun1 3653 . . . . . . . . . . 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 2514 . . . . . . . . . 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 2909 . . . . . . . 8  |-  ( E. f  e.  C  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) )  ->  E. f  e.  C  x  e.  dom  f )
3635ralimi 2834 . . . . . . 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 4316 . . . . . . 7  |-  ( x  e.  U_ f  e.  C  dom  f  <->  E. f  e.  C  x  e.  dom  f )
3938ralbii 2872 . . . . . 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 2603 . . . . . 6  |-  F/_ x A
428bnj1309 33785 . . . . . . . . 9  |-  ( t  e.  B  ->  A. x  t  e.  B )
432, 42bnj1307 33786 . . . . . . . 8  |-  ( t  e.  C  ->  A. x  t  e.  C )
4443nfcii 2593 . . . . . . 7  |-  F/_ x C
45 nfcv 2603 . . . . . . 7  |-  F/_ x dom  f
4644, 45nfiun 4339 . . . . . 6  |-  F/_ x U_ f  e.  C  dom  f
4741, 46dfss3f 3478 . . . . 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 3533 . . 3  |-  ( R 
FrSe  A  ->  A  C_  dom  U. C )
5049, 24syl6sseqr 3533 . 2  |-  ( R 
FrSe  A  ->  A  C_  dom  F )
5127, 50eqssd 3503 1  |-  ( R 
FrSe  A  ->  dom  F  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802   {cab 2426   A.wral 2791   E.wrex 2792    u. cun 3456    C_ wss 3458   {csn 4010   <.cop 4016   U.cuni 4230   U_ciun 4311   dom cdm 4985    |` cres 4987    Fn wfn 5569   ` cfv 5574    predc-bnj14 33448    FrSe w-bnj15 33452    trClc-bnj18 33454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-reg 8016  ax-inf2 8056
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-om 6682  df-1o 7128  df-bnj17 33447  df-bnj14 33449  df-bnj13 33451  df-bnj15 33453  df-bnj18 33455  df-bnj19 33457
This theorem is referenced by:  bnj60  33825
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