MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2xopoveqd Structured version   Unicode version

Theorem mpt2xopoveqd 6975
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, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
Hypotheses
Ref Expression
mpt2xopoveq.f  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )
mpt2xopoveqd.1  |-  ( ps 
->  ( V  e.  X  /\  W  e.  Y
) )
mpt2xopoveqd.2  |-  ( ( ps  /\  -.  K  e.  V )  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  =  (/) )
Assertion
Ref Expression
mpt2xopoveqd  |-  ( ps 
->  ( <. 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    x, F
Allowed substitution hints:    ph( x, y, n)    ps( x, y, n)    F( y, n)

Proof of Theorem mpt2xopoveqd
StepHypRef Expression
1 mpt2xopoveqd.1 . . . 4  |-  ( ps 
->  ( V  e.  X  /\  W  e.  Y
) )
2 mpt2xopoveq.f . . . . . 6  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )
32mpt2xopoveq 6973 . . . . 5  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( <. V ,  W >. F K )  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
43ex 435 . . . 4  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( K  e.  V  ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
) )
51, 4syl 17 . . 3  |-  ( ps 
->  ( K  e.  V  ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
) )
65com12 32 . 2  |-  ( K  e.  V  ->  ( ps  ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph } ) )
7 df-nel 2628 . . . . . 6  |-  ( K  e/  V  <->  -.  K  e.  V )
82mpt2xopynvov0 6972 . . . . . 6  |-  ( K  e/  V  ->  ( <. V ,  W >. F K )  =  (/) )
97, 8sylbir 216 . . . . 5  |-  ( -.  K  e.  V  -> 
( <. V ,  W >. F K )  =  (/) )
109adantr 466 . . . 4  |-  ( ( -.  K  e.  V  /\  ps )  ->  ( <. V ,  W >. F K )  =  (/) )
11 mpt2xopoveqd.2 . . . . . 6  |-  ( ( ps  /\  -.  K  e.  V )  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  =  (/) )
1211eqcomd 2437 . . . . 5  |-  ( ( ps  /\  -.  K  e.  V )  ->  (/)  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
1312ancoms 454 . . . 4  |-  ( ( -.  K  e.  V  /\  ps )  ->  (/)  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
1410, 13eqtrd 2470 . . 3  |-  ( ( -.  K  e.  V  /\  ps )  ->  ( <. V ,  W >. F K )  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
1514ex 435 . 2  |-  ( -.  K  e.  V  -> 
( ps  ->  ( <. V ,  W >. F K )  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
) )
166, 15pm2.61i 167 1  |-  ( ps 
->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870    e/ wnel 2626   {crab 2786   _Vcvv 3087   [.wsbc 3305   (/)c0 3767   <.cop 4008   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1stc1st 6805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator