ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mpt2xopoveq Unicode version

Theorem mpt2xopoveq 5886
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 5206 . . . . 5  |-  ( x  =  <. V ,  W >.  ->  ( 1st `  x
)  =  ( 1st `  <. V ,  W >. ) )
4 op1stg 5805 . . . . . 6  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( 1st `  <. V ,  W >. )  =  V )
54adantr 265 . . . . 5  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( 1st `  <. V ,  W >. )  =  V )
63, 5sylan9eqr 2110 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  x  =  <. V ,  W >. )  ->  ( 1st `  x )  =  V )
76adantrr 456 . . 3  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( 1st `  x
)  =  V )
8 sbceq1a 2796 . . . . . 6  |-  ( y  =  K  ->  ( ph 
<-> 
[. K  /  y ]. ph ) )
98adantl 266 . . . . 5  |-  ( ( x  =  <. V ,  W >.  /\  y  =  K )  ->  ( ph 
<-> 
[. K  /  y ]. ph ) )
109adantl 266 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( ph  <->  [. K  /  y ]. ph ) )
11 sbceq1a 2796 . . . . . 6  |-  ( x  =  <. V ,  W >.  ->  ( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph ) )
1211adantr 265 . . . . 5  |-  ( ( x  =  <. V ,  W >.  /\  y  =  K )  ->  ( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
1312adantl 266 . . . 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 181 . . 3  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
157, 14rabeqbidv 2569 . 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 3992 . . 3  |-  ( ( V  e.  X  /\  W  e.  Y )  -> 
<. V ,  W >.  e. 
_V )
1716adantr 265 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  <. V ,  W >.  e.  _V )
18 simpr 107 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  K  e.  V )
19 rabexg 3928 . . 3  |-  ( V  e.  X  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  e.  _V )
2019ad2antrr 465 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  e.  _V )
21 equid 1605 . . 3  |-  z  =  z
22 nfvd 1438 . . 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 1438 . . 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 2194 . 2  |-  F/_ y <. V ,  W >.
27 nfcv 2194 . 2  |-  F/_ x K
28 nfsbc1v 2805 . . 3  |-  F/ x [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph
29 nfcv 2194 . . 3  |-  F/_ x V
3028, 29nfrabxy 2507 . 2  |-  F/_ x { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
31 nfsbc1v 2805 . . . 4  |-  F/ y
[. K  /  y ]. ph
3226, 31nfsbc 2807 . . 3  |-  F/ y
[. <. V ,  W >.  /  x ]. [. K  /  y ]. ph
33 nfcv 2194 . . 3  |-  F/_ y V
3432, 33nfrabxy 2507 . 2  |-  F/_ y { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpt2dxf 5654 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 101    <-> wb 102    = wceq 1259   F/wnf 1365    e. wcel 1409   {crab 2327   _Vcvv 2574   [.wsbc 2787   <.cop 3406   ` cfv 4930  (class class class)co 5540    |-> cmpt2 5542   1stc1st 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795
This theorem is referenced by:  mpt2xopovel  5887
  Copyright terms: Public domain W3C validator