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Theorem bnj1498 34464
Description: Technical lemma for bnj60 34465. 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 4248 . . . . . . 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 34245 . . . . . . . . . . . . . . 15  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
43bnj1299 34224 . . . . . . . . . . . . . 14  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
5 fndm 5588 . . . . . . . . . . . . . 14  |-  ( f  Fn  d  ->  dom  f  =  d )
64, 5bnj31 34119 . . . . . . . . . . . . 13  |-  ( f  e.  C  ->  E. d  e.  B  dom  f  =  d )
76bnj1196 34200 . . . . . . . . . . . 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 34245 . . . . . . . . . . . . . 14  |-  ( d  e.  B  ->  (
d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) )
109simpld 457 . . . . . . . . . . . . 13  |-  ( d  e.  B  ->  d  C_  A )
1110anim1i 566 . . . . . . . . . . . 12  |-  ( ( d  e.  B  /\  dom  f  =  d
)  ->  ( d  C_  A  /\  dom  f  =  d ) )
127, 11bnj593 34149 . . . . . . . . . . 11  |-  ( f  e.  C  ->  E. d
( d  C_  A  /\  dom  f  =  d ) )
13 sseq1 3438 . . . . . . . . . . . 12  |-  ( dom  f  =  d  -> 
( dom  f  C_  A 
<->  d  C_  A )
)
1413biimparc 485 . . . . . . . . . . 11  |-  ( ( d  C_  A  /\  dom  f  =  d
)  ->  dom  f  C_  A )
1512, 14bnj593 34149 . . . . . . . . . 10  |-  ( f  e.  C  ->  E. d dom  f  C_  A )
1615bnj937 34177 . . . . . . . . 9  |-  ( f  e.  C  ->  dom  f  C_  A )
1716sselda 3417 . . . . . . . 8  |-  ( ( f  e.  C  /\  z  e.  dom  f )  ->  z  e.  A
)
1817rexlimiva 2870 . . . . . . 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 34227 . . . . . . 7  |-  ( w  e.  C  ->  A. f  w  e.  C )
2120bnj1400 34241 . . . . . 6  |-  dom  U. C  =  U_ f  e.  C  dom  f
2219, 21eleq2s 2490 . . . . 5  |-  ( z  e.  dom  U. C  ->  z  e.  A )
23 bnj1498.4 . . . . . 6  |-  F  = 
U. C
2423dmeqi 5117 . . . . 5  |-  dom  F  =  dom  U. C
2522, 24eleq2s 2490 . . . 4  |-  ( z  e.  dom  F  -> 
z  e.  A )
2625ssriv 3421 . . 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 34462 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
30 ssnid 3973 . . . . . . . . . . 11  |-  x  e. 
{ x }
31 elun1 3585 . . . . . . . . . . 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 2455 . . . . . . . . . 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 2850 . . . . . . . 8  |-  ( E. f  e.  C  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) )  ->  E. f  e.  C  x  e.  dom  f )
3635ralimi 2775 . . . . . . 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 4248 . . . . . . 7  |-  ( x  e.  U_ f  e.  C  dom  f  <->  E. f  e.  C  x  e.  dom  f )
3938ralbii 2813 . . . . . 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 2544 . . . . . 6  |-  F/_ x A
428bnj1309 34425 . . . . . . . . 9  |-  ( t  e.  B  ->  A. x  t  e.  B )
432, 42bnj1307 34426 . . . . . . . 8  |-  ( t  e.  C  ->  A. x  t  e.  C )
4443nfcii 2534 . . . . . . 7  |-  F/_ x C
45 nfcv 2544 . . . . . . 7  |-  F/_ x dom  f
4644, 45nfiun 4271 . . . . . 6  |-  F/_ x U_ f  e.  C  dom  f
4741, 46dfss3f 3409 . . . . 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 3464 . . 3  |-  ( R 
FrSe  A  ->  A  C_  dom  U. C )
5049, 24syl6sseqr 3464 . 2  |-  ( R 
FrSe  A  ->  A  C_  dom  F )
5127, 50eqssd 3434 1  |-  ( R 
FrSe  A  ->  dom  F  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   {cab 2367   A.wral 2732   E.wrex 2733    u. cun 3387    C_ wss 3389   {csn 3944   <.cop 3950   U.cuni 4163   U_ciun 4243   dom cdm 4913    |` cres 4915    Fn wfn 5491   ` cfv 5496    predc-bnj14 34087    FrSe w-bnj15 34091    trClc-bnj18 34093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-reg 7933  ax-inf2 7972
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-om 6600  df-1o 7048  df-bnj17 34086  df-bnj14 34088  df-bnj13 34090  df-bnj15 34092  df-bnj18 34094  df-bnj19 34096
This theorem is referenced by:  bnj60  34465
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