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Theorem mpt2xopoveq 6005
Description: Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
Hypothesis
Ref Expression
mpt2xopoveq.f  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )
Assertion
Ref Expression
mpt2xopoveq  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( <. V ,  W >. F K )  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
Distinct variable groups:    n, K, x, y    n, V, x, y    n, W, x, y    n, X, x, y    n, Y, x, y
Allowed substitution hints:    ph( x, y, n)    F( x, y, n)

Proof of Theorem mpt2xopoveq
StepHypRef Expression
1 mpt2xopoveq.f . . 3  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )
21a1i 9 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  F  =  ( x  e. 
_V ,  y  e.  ( 1st `  x
)  |->  { n  e.  ( 1st `  x
)  |  ph }
) )
3 fveq2 5305 . . . . 5  |-  ( x  =  <. V ,  W >.  ->  ( 1st `  x
)  =  ( 1st `  <. V ,  W >. ) )
4 op1stg 5921 . . . . . 6  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( 1st `  <. V ,  W >. )  =  V )
54adantr 270 . . . . 5  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( 1st `  <. V ,  W >. )  =  V )
63, 5sylan9eqr 2142 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  x  =  <. V ,  W >. )  ->  ( 1st `  x )  =  V )
76adantrr 463 . . 3  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( 1st `  x
)  =  V )
8 sbceq1a 2849 . . . . . 6  |-  ( y  =  K  ->  ( ph 
<-> 
[. K  /  y ]. ph ) )
98adantl 271 . . . . 5  |-  ( ( x  =  <. V ,  W >.  /\  y  =  K )  ->  ( ph 
<-> 
[. K  /  y ]. ph ) )
109adantl 271 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( ph  <->  [. K  /  y ]. ph ) )
11 sbceq1a 2849 . . . . . 6  |-  ( x  =  <. V ,  W >.  ->  ( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph ) )
1211adantr 270 . . . . 5  |-  ( ( x  =  <. V ,  W >.  /\  y  =  K )  ->  ( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
1312adantl 271 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
1410, 13bitrd 186 . . 3  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
157, 14rabeqbidv 2614 . 2  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  ->  { n  e.  ( 1st `  x )  | 
ph }  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
16 opexg 4055 . . 3  |-  ( ( V  e.  X  /\  W  e.  Y )  -> 
<. V ,  W >.  e. 
_V )
1716adantr 270 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  <. V ,  W >.  e.  _V )
18 simpr 108 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  K  e.  V )
19 rabexg 3982 . . 3  |-  ( V  e.  X  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  e.  _V )
2019ad2antrr 472 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  e.  _V )
21 equid 1634 . . 3  |-  z  =  z
22 nfvd 1467 . . 3  |-  ( z  =  z  ->  F/ x ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V ) )
2321, 22ax-mp 7 . 2  |-  F/ x
( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )
24 nfvd 1467 . . 3  |-  ( z  =  z  ->  F/ y ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V ) )
2521, 24ax-mp 7 . 2  |-  F/ y ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )
26 nfcv 2228 . 2  |-  F/_ y <. V ,  W >.
27 nfcv 2228 . 2  |-  F/_ x K
28 nfsbc1v 2858 . . 3  |-  F/ x [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph
29 nfcv 2228 . . 3  |-  F/_ x V
3028, 29nfrabxy 2547 . 2  |-  F/_ x { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
31 nfsbc1v 2858 . . . 4  |-  F/ y
[. K  /  y ]. ph
3226, 31nfsbc 2860 . . 3  |-  F/ y
[. <. V ,  W >.  /  x ]. [. K  /  y ]. ph
33 nfcv 2228 . . 3  |-  F/_ y V
3432, 33nfrabxy 2547 . 2  |-  F/_ y { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpt2dxf 5770 1  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( <. V ,  W >. F K )  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   F/wnf 1394    e. wcel 1438   {crab 2363   _Vcvv 2619   [.wsbc 2840   <.cop 3449   ` cfv 5015  (class class class)co 5652    |-> cmpt2 5654   1stc1st 5909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911
This theorem is referenced by:  mpt2xopovel  6006
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