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Theorem bnj1498 29407
Description: Technical lemma for bnj60 29408. 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 3925 . . . . . . 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 29188 . . . . . . . . . . . . . . 15  |-  ( f  e.  C  ->  E. d  e.  B  ( f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) )
43bnj1299 29167 . . . . . . . . . . . . . 14  |-  ( f  e.  C  ->  E. d  e.  B  f  Fn  d )
5 fndm 5359 . . . . . . . . . . . . . 14  |-  ( f  Fn  d  ->  dom  f  =  d )
64, 5bnj31 29061 . . . . . . . . . . . . 13  |-  ( f  e.  C  ->  E. d  e.  B  dom  f  =  d )
76bnj1196 29143 . . . . . . . . . . . 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 29188 . . . . . . . . . . . . . 14  |-  ( d  e.  B  ->  (
d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) )
109simpld 445 . . . . . . . . . . . . 13  |-  ( d  e.  B  ->  d  C_  A )
1110anim1i 551 . . . . . . . . . . . 12  |-  ( ( d  e.  B  /\  dom  f  =  d
)  ->  ( d  C_  A  /\  dom  f  =  d ) )
127, 11bnj593 29090 . . . . . . . . . . 11  |-  ( f  e.  C  ->  E. d
( d  C_  A  /\  dom  f  =  d ) )
13 sseq1 3212 . . . . . . . . . . . 12  |-  ( dom  f  =  d  -> 
( dom  f  C_  A 
<->  d  C_  A )
)
1413biimparc 473 . . . . . . . . . . 11  |-  ( ( d  C_  A  /\  dom  f  =  d
)  ->  dom  f  C_  A )
1512, 14bnj593 29090 . . . . . . . . . 10  |-  ( f  e.  C  ->  E. d dom  f  C_  A )
1615bnj937 29119 . . . . . . . . 9  |-  ( f  e.  C  ->  dom  f  C_  A )
1716sselda 3193 . . . . . . . 8  |-  ( ( f  e.  C  /\  z  e.  dom  f )  ->  z  e.  A
)
1817rexlimiva 2675 . . . . . . 7  |-  ( E. f  e.  C  z  e.  dom  f  -> 
z  e.  A )
191, 18sylbi 187 . . . . . 6  |-  ( z  e.  U_ f  e.  C  dom  f  -> 
z  e.  A )
202bnj1317 29170 . . . . . . 7  |-  ( w  e.  C  ->  A. f  w  e.  C )
2120bnj1400 29184 . . . . . 6  |-  dom  U. C  =  U_ f  e.  C  dom  f
2219, 21eleq2s 2388 . . . . 5  |-  ( z  e.  dom  U. C  ->  z  e.  A )
23 bnj1498.4 . . . . . 6  |-  F  = 
U. C
2423dmeqi 4896 . . . . 5  |-  dom  F  =  dom  U. C
2522, 24eleq2s 2388 . . . 4  |-  ( z  e.  dom  F  -> 
z  e.  A )
2625ssriv 3197 . . 3  |-  dom  F  C_  A
2726a1i 10 . 2  |-  ( R 
FrSe  A  ->  dom  F  C_  A )
28 bnj1498.2 . . . . . . . 8  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
298, 28, 2bnj1493 29405 . . . . . . 7  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) ) )
30 vex 2804 . . . . . . . . . . . 12  |-  x  e. 
_V
3130snid 3680 . . . . . . . . . . 11  |-  x  e. 
{ x }
32 elun1 3355 . . . . . . . . . . 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 2357 . . . . . . . . . 10  |-  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  -> 
( x  e.  dom  f 
<->  x  e.  ( { x }  u.  trCl ( x ,  A ,  R ) ) ) )
3533, 34mpbiri 224 . . . . . . . . 9  |-  ( dom  f  =  ( { x }  u.  trCl ( x ,  A ,  R ) )  ->  x  e.  dom  f )
3635reximi 2663 . . . . . . . 8  |-  ( E. f  e.  C  dom  f  =  ( {
x }  u.  trCl ( x ,  A ,  R ) )  ->  E. f  e.  C  x  e.  dom  f )
3736ralimi 2631 . . . . . . 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 15 . . . . . 6  |-  ( R 
FrSe  A  ->  A. x  e.  A  E. f  e.  C  x  e.  dom  f )
39 eliun 3925 . . . . . . 7  |-  ( x  e.  U_ f  e.  C  dom  f  <->  E. f  e.  C  x  e.  dom  f )
4039ralbii 2580 . . . . . 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 203 . . . . 5  |-  ( R 
FrSe  A  ->  A. x  e.  A  x  e.  U_ f  e.  C  dom  f )
42 nfcv 2432 . . . . . 6  |-  F/_ x A
438bnj1309 29368 . . . . . . . . 9  |-  ( t  e.  B  ->  A. x  t  e.  B )
442, 43bnj1307 29369 . . . . . . . 8  |-  ( t  e.  C  ->  A. x  t  e.  C )
4544nfcii 2423 . . . . . . 7  |-  F/_ x C
46 nfcv 2432 . . . . . . 7  |-  F/_ x dom  f
4745, 46nfiun 3947 . . . . . 6  |-  F/_ x U_ f  e.  C  dom  f
4842, 47dfss3f 3185 . . . . 5  |-  ( A 
C_  U_ f  e.  C  dom  f  <->  A. x  e.  A  x  e.  U_ f  e.  C  dom  f )
4941, 48sylibr 203 . . . 4  |-  ( R 
FrSe  A  ->  A  C_  U_ f  e.  C  dom  f )
5049, 21syl6sseqr 3238 . . 3  |-  ( R 
FrSe  A  ->  A  C_  dom  U. C )
5150, 24syl6sseqr 3238 . 2  |-  ( R 
FrSe  A  ->  A  C_  dom  F )
5227, 51eqssd 3209 1  |-  ( R 
FrSe  A  ->  dom  F  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    u. cun 3163    C_ wss 3165   {csn 3653   <.cop 3656   U.cuni 3843   U_ciun 3921   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271    predc-bnj14 29029    FrSe w-bnj15 29033    trClc-bnj18 29035
This theorem is referenced by:  bnj60  29408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-bnj17 29028  df-bnj14 29030  df-bnj13 29032  df-bnj15 29034  df-bnj18 29036  df-bnj19 29038
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