Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mpt2xopovel Unicode version

Theorem mpt2xopovel 27260
Description: Element of the 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 and Mario Carneiro, 10-Oct-2017.)
Hypothesis
Ref Expression
mpt2xopoveq.f  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )
Assertion
Ref Expression
mpt2xopovel  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  (
<. V ,  W >. F K )  <->  ( K  e.  V  /\  N  e.  V  /\  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph ) ) )
Distinct variable groups:    n, K, x, y    n, V, x, y    n, W, x, y    n, X, x, y    n, Y, x, y    x, N, y
Allowed substitution hints:    ph( x, y, n)    F( x, y, n)    N( n)

Proof of Theorem mpt2xopovel
StepHypRef Expression
1 mpt2xopoveq.f . . . 4  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )
21mpt2xopn0yelv 27253 . . 3  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  (
<. V ,  W >. F K )  ->  K  e.  V ) )
32pm4.71rd 616 . 2  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  (
<. V ,  W >. F K )  <->  ( K  e.  V  /\  N  e.  ( <. V ,  W >. F K ) ) ) )
41mpt2xopoveq 27259 . . . . . 6  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( <. V ,  W >. F K )  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
54eleq2d 2383 . . . . 5  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( N  e.  ( <. V ,  W >. F K )  <->  N  e.  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph } ) )
6 nfcv 2452 . . . . . . 7  |-  F/_ n V
76elrabsf 3063 . . . . . 6  |-  ( N  e.  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  <->  ( N  e.  V  /\  [. N  /  n ]. [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph ) )
8 sbccom 3096 . . . . . . . 8  |-  ( [. N  /  n ]. [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. N  /  n ]. [. K  /  y ]. ph )
9 sbccom 3096 . . . . . . . . 9  |-  ( [. N  /  n ]. [. K  /  y ]. ph  <->  [. K  / 
y ]. [. N  /  n ]. ph )
109sbcbii 3080 . . . . . . . 8  |-  ( [. <. V ,  W >.  /  x ]. [. N  /  n ]. [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph )
118, 10bitri 240 . . . . . . 7  |-  ( [. N  /  n ]. [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph )
1211anbi2i 675 . . . . . 6  |-  ( ( N  e.  V  /\  [. N  /  n ]. [.
<. V ,  W >.  /  x ]. [. K  /  y ]. ph )  <->  ( N  e.  V  /\  [.
<. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph )
)
137, 12bitri 240 . . . . 5  |-  ( N  e.  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  <->  ( N  e.  V  /\  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph ) )
145, 13syl6bb 252 . . . 4  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( N  e.  ( <. V ,  W >. F K )  <->  ( N  e.  V  /\  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph ) ) )
1514pm5.32da 622 . . 3  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( ( K  e.  V  /\  N  e.  ( <. V ,  W >. F K ) )  <-> 
( K  e.  V  /\  ( N  e.  V  /\  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph )
) ) )
16 3anass 938 . . 3  |-  ( ( K  e.  V  /\  N  e.  V  /\  [.
<. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph )  <->  ( K  e.  V  /\  ( N  e.  V  /\  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph )
) )
1715, 16syl6bbr 254 . 2  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( ( K  e.  V  /\  N  e.  ( <. V ,  W >. F K ) )  <-> 
( K  e.  V  /\  N  e.  V  /\  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph )
) )
183, 17bitrd 244 1  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( N  e.  (
<. V ,  W >. F K )  <->  ( K  e.  V  /\  N  e.  V  /\  [. <. V ,  W >.  /  x ]. [. K  /  y ]. [. N  /  n ]. ph ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   {crab 2581   _Vcvv 2822   [.wsbc 3025   <.cop 3677   ` cfv 5292  (class class class)co 5900    e. cmpt2 5902   1stc1st 6162
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165
  Copyright terms: Public domain W3C validator