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Theorem aciunf1 27933
Description: Choice in an index union. (Contributed by Thierry Arnoux, 4-May-2020.)
Hypotheses
Ref Expression
aciunf1.0  |-  ( ph  ->  A  e.  V )
aciunf1.1  |-  ( (
ph  /\  j  e.  A )  ->  B  e.  W )
Assertion
Ref Expression
aciunf1  |-  ( ph  ->  E. f ( f : U_ j  e.  A  B -1-1-> U_ j  e.  A  ( {
j }  X.  B
)  /\  A. k  e.  U_  j  e.  A  B ( 2nd `  (
f `  k )
)  =  k ) )
Distinct variable groups:    A, j,
k, f    B, f,
k    j, W    ph, f, j, k
Allowed substitution hints:    B( j)    V( f, j, k)    W( f, k)

Proof of Theorem aciunf1
StepHypRef Expression
1 ssrab2 3523 . . . 4  |-  { j  e.  A  |  B  =/=  (/) }  C_  A
2 aciunf1.0 . . . 4  |-  ( ph  ->  A  e.  V )
3 ssexg 4539 . . . 4  |-  ( ( { j  e.  A  |  B  =/=  (/) }  C_  A  /\  A  e.  V
)  ->  { j  e.  A  |  B  =/=  (/) }  e.  _V )
41, 2, 3sylancr 661 . . 3  |-  ( ph  ->  { j  e.  A  |  B  =/=  (/) }  e.  _V )
5 rabid 2983 . . . . . 6  |-  ( j  e.  { j  e.  A  |  B  =/=  (/) }  <->  ( j  e.  A  /\  B  =/=  (/) ) )
65biimpi 194 . . . . 5  |-  ( j  e.  { j  e.  A  |  B  =/=  (/) }  ->  ( j  e.  A  /\  B  =/=  (/) ) )
76adantl 464 . . . 4  |-  ( (
ph  /\  j  e.  { j  e.  A  |  B  =/=  (/) } )  -> 
( j  e.  A  /\  B  =/=  (/) ) )
87simprd 461 . . 3  |-  ( (
ph  /\  j  e.  { j  e.  A  |  B  =/=  (/) } )  ->  B  =/=  (/) )
9 nfrab1 2987 . . 3  |-  F/_ j { j  e.  A  |  B  =/=  (/) }
107simpld 457 . . . 4  |-  ( (
ph  /\  j  e.  { j  e.  A  |  B  =/=  (/) } )  -> 
j  e.  A )
11 aciunf1.1 . . . 4  |-  ( (
ph  /\  j  e.  A )  ->  B  e.  W )
1210, 11syldan 468 . . 3  |-  ( (
ph  /\  j  e.  { j  e.  A  |  B  =/=  (/) } )  ->  B  e.  W )
134, 8, 9, 12aciunf1lem 27932 . 2  |-  ( ph  ->  E. f ( f : U_ j  e. 
{ j  e.  A  |  B  =/=  (/) } B -1-1-> U_ j  e.  { j  e.  A  |  B  =/=  (/) }  ( { j }  X.  B
)  /\  A. k  e.  U_  j  e.  {
j  e.  A  |  B  =/=  (/) } B ( 2nd `  ( f `
 k ) )  =  k ) )
14 eqidd 2403 . . . . 5  |-  ( ph  ->  f  =  f )
15 nfv 1728 . . . . . . 7  |-  F/ j
ph
16 nfcv 2564 . . . . . . . 8  |-  F/_ j A
17 nfrab1 2987 . . . . . . . 8  |-  F/_ j { j  e.  A  |  B  =  (/) }
1816, 17nfdif 3563 . . . . . . 7  |-  F/_ j
( A  \  {
j  e.  A  |  B  =  (/) } )
19 difrab 3723 . . . . . . . . 9  |-  ( { j  e.  A  | T.  }  \  { j  e.  A  |  B  =  (/) } )  =  { j  e.  A  |  ( T.  /\  -.  B  =  (/) ) }
2016rabtru 27800 . . . . . . . . . 10  |-  { j  e.  A  | T.  }  =  A
2120difeq1i 3556 . . . . . . . . 9  |-  ( { j  e.  A  | T.  }  \  { j  e.  A  |  B  =  (/) } )  =  ( A  \  {
j  e.  A  |  B  =  (/) } )
22 truan 1422 . . . . . . . . . . . . 13  |-  ( ( T.  /\  -.  B  =  (/) )  <->  -.  B  =  (/) )
23 df-ne 2600 . . . . . . . . . . . . 13  |-  ( B  =/=  (/)  <->  -.  B  =  (/) )
2422, 23bitr4i 252 . . . . . . . . . . . 12  |-  ( ( T.  /\  -.  B  =  (/) )  <->  B  =/=  (/) )
2524a1i 11 . . . . . . . . . . 11  |-  ( T. 
->  ( ( T.  /\  -.  B  =  (/) )  <->  B  =/=  (/) ) )
2625rabbidv 3050 . . . . . . . . . 10  |-  ( T. 
->  { j  e.  A  |  ( T.  /\  -.  B  =  (/) ) }  =  { j  e.  A  |  B  =/=  (/) } )
2726trud 1414 . . . . . . . . 9  |-  { j  e.  A  |  ( T.  /\  -.  B  =  (/) ) }  =  { j  e.  A  |  B  =/=  (/) }
2819, 21, 273eqtr3i 2439 . . . . . . . 8  |-  ( A 
\  { j  e.  A  |  B  =  (/) } )  =  {
j  e.  A  |  B  =/=  (/) }
2928a1i 11 . . . . . . 7  |-  ( ph  ->  ( A  \  {
j  e.  A  |  B  =  (/) } )  =  { j  e.  A  |  B  =/=  (/) } )
30 eqidd 2403 . . . . . . 7  |-  ( ph  ->  B  =  B )
3115, 18, 9, 29, 30iuneq12df 4294 . . . . . 6  |-  ( ph  ->  U_ j  e.  ( A  \  { j  e.  A  |  B  =  (/) } ) B  =  U_ j  e. 
{ j  e.  A  |  B  =/=  (/) } B
)
32 rabid 2983 . . . . . . . . . . 11  |-  ( j  e.  { j  e.  A  |  B  =  (/) }  <->  ( j  e.  A  /\  B  =  (/) ) )
3332biimpi 194 . . . . . . . . . 10  |-  ( j  e.  { j  e.  A  |  B  =  (/) }  ->  ( j  e.  A  /\  B  =  (/) ) )
3433adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  { j  e.  A  |  B  =  (/) } )  ->  ( j  e.  A  /\  B  =  (/) ) )
3534simprd 461 . . . . . . . 8  |-  ( (
ph  /\  j  e.  { j  e.  A  |  B  =  (/) } )  ->  B  =  (/) )
3635ralrimiva 2817 . . . . . . 7  |-  ( ph  ->  A. j  e.  {
j  e.  A  |  B  =  (/) } B  =  (/) )
3717iunxdif3 27843 . . . . . . 7  |-  ( A. j  e.  { j  e.  A  |  B  =  (/) } B  =  (/)  ->  U_ j  e.  ( A  \  { j  e.  A  |  B  =  (/) } ) B  =  U_ j  e.  A  B )
3836, 37syl 17 . . . . . 6  |-  ( ph  ->  U_ j  e.  ( A  \  { j  e.  A  |  B  =  (/) } ) B  =  U_ j  e.  A  B )
3931, 38eqtr3d 2445 . . . . 5  |-  ( ph  ->  U_ j  e.  {
j  e.  A  |  B  =/=  (/) } B  = 
U_ j  e.  A  B )
40 eqidd 2403 . . . . . . 7  |-  ( ph  ->  ( { j }  X.  B )  =  ( { j }  X.  B ) )
4115, 18, 9, 29, 40iuneq12df 4294 . . . . . 6  |-  ( ph  ->  U_ j  e.  ( A  \  { j  e.  A  |  B  =  (/) } ) ( { j }  X.  B )  =  U_ j  e.  { j  e.  A  |  B  =/=  (/) }  ( { j }  X.  B
) )
4235xpeq2d 4846 . . . . . . . . 9  |-  ( (
ph  /\  j  e.  { j  e.  A  |  B  =  (/) } )  ->  ( { j }  X.  B )  =  ( { j }  X.  (/) ) )
43 xp0 5242 . . . . . . . . 9  |-  ( { j }  X.  (/) )  =  (/)
4442, 43syl6eq 2459 . . . . . . . 8  |-  ( (
ph  /\  j  e.  { j  e.  A  |  B  =  (/) } )  ->  ( { j }  X.  B )  =  (/) )
4544ralrimiva 2817 . . . . . . 7  |-  ( ph  ->  A. j  e.  {
j  e.  A  |  B  =  (/) }  ( { j }  X.  B )  =  (/) )
4617iunxdif3 27843 . . . . . . 7  |-  ( A. j  e.  { j  e.  A  |  B  =  (/) }  ( { j }  X.  B
)  =  (/)  ->  U_ j  e.  ( A  \  {
j  e.  A  |  B  =  (/) } ) ( { j }  X.  B )  = 
U_ j  e.  A  ( { j }  X.  B ) )
4745, 46syl 17 . . . . . 6  |-  ( ph  ->  U_ j  e.  ( A  \  { j  e.  A  |  B  =  (/) } ) ( { j }  X.  B )  =  U_ j  e.  A  ( { j }  X.  B ) )
4841, 47eqtr3d 2445 . . . . 5  |-  ( ph  ->  U_ j  e.  {
j  e.  A  |  B  =/=  (/) }  ( { j }  X.  B
)  =  U_ j  e.  A  ( {
j }  X.  B
) )
4914, 39, 48f1eq123d 5793 . . . 4  |-  ( ph  ->  ( f : U_ j  e.  { j  e.  A  |  B  =/=  (/) } B -1-1-> U_ j  e.  { j  e.  A  |  B  =/=  (/) }  ( { j }  X.  B
)  <->  f : U_ j  e.  A  B -1-1-> U_ j  e.  A  ( { j }  X.  B ) ) )
5039raleqdv 3009 . . . 4  |-  ( ph  ->  ( A. k  e. 
U_  j  e.  {
j  e.  A  |  B  =/=  (/) } B ( 2nd `  ( f `
 k ) )  =  k  <->  A. k  e.  U_  j  e.  A  B ( 2nd `  (
f `  k )
)  =  k ) )
5149, 50anbi12d 709 . . 3  |-  ( ph  ->  ( ( f :
U_ j  e.  {
j  e.  A  |  B  =/=  (/) } B -1-1-> U_ j  e.  { j  e.  A  |  B  =/=  (/) }  ( { j }  X.  B
)  /\  A. k  e.  U_  j  e.  {
j  e.  A  |  B  =/=  (/) } B ( 2nd `  ( f `
 k ) )  =  k )  <->  ( f : U_ j  e.  A  B -1-1-> U_ j  e.  A  ( { j }  X.  B )  /\  A. k  e.  U_  j  e.  A  B ( 2nd `  ( f `  k
) )  =  k ) ) )
5251exbidv 1735 . 2  |-  ( ph  ->  ( E. f ( f : U_ j  e.  { j  e.  A  |  B  =/=  (/) } B -1-1-> U_ j  e.  { j  e.  A  |  B  =/=  (/) }  ( { j }  X.  B
)  /\  A. k  e.  U_  j  e.  {
j  e.  A  |  B  =/=  (/) } B ( 2nd `  ( f `
 k ) )  =  k )  <->  E. f
( f : U_ j  e.  A  B -1-1-> U_ j  e.  A  ( { j }  X.  B )  /\  A. k  e.  U_  j  e.  A  B ( 2nd `  ( f `  k
) )  =  k ) ) )
5313, 52mpbid 210 1  |-  ( ph  ->  E. f ( f : U_ j  e.  A  B -1-1-> U_ j  e.  A  ( {
j }  X.  B
)  /\  A. k  e.  U_  j  e.  A  B ( 2nd `  (
f `  k )
)  =  k ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   T. wtru 1406   E.wex 1633    e. wcel 1842    =/= wne 2598   A.wral 2753   {crab 2757   _Vcvv 3058    \ cdif 3410    C_ wss 3413   (/)c0 3737   {csn 3971   U_ciun 4270    X. cxp 4820   -1-1->wf1 5565   ` cfv 5568   2ndc2nd 6782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-reg 8051  ax-inf2 8090  ax-ac2 8874
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-isom 5577  df-riota 6239  df-om 6683  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-en 7554  df-r1 8213  df-rank 8214  df-card 8351  df-ac 8528
This theorem is referenced by:  esumiun  28527
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