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Theorem mpt2xopoveqd 6949
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 6947 . . . . 5  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( <. V ,  W >. F K )  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
43ex 434 . . . 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 16 . . 3  |-  ( ps 
->  ( K  e.  V  ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
) )
65com12 31 . 2  |-  ( K  e.  V  ->  ( ps  ->  ( <. V ,  W >. F K )  =  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph } ) )
7 df-nel 2665 . . . . . 6  |-  ( K  e/  V  <->  -.  K  e.  V )
82mpt2xopynvov0 6946 . . . . . 6  |-  ( K  e/  V  ->  ( <. V ,  W >. F K )  =  (/) )
97, 8sylbir 213 . . . . 5  |-  ( -.  K  e.  V  -> 
( <. V ,  W >. F K )  =  (/) )
109adantr 465 . . . 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 2475 . . . . 5  |-  ( ( ps  /\  -.  K  e.  V )  ->  (/)  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
1312ancoms 453 . . . 4  |-  ( ( -.  K  e.  V  /\  ps )  ->  (/)  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
1410, 13eqtrd 2508 . . 3  |-  ( ( -.  K  e.  V  /\  ps )  ->  ( <. V ,  W >. F K )  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
)
1514ex 434 . 2  |-  ( -.  K  e.  V  -> 
( ps  ->  ( <. V ,  W >. F K )  =  {
n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
) )
166, 15pm2.61i 164 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 369    = wceq 1379    e. wcel 1767    e/ wnel 2663   {crab 2818   _Vcvv 3113   [.wsbc 3331   (/)c0 3785   <.cop 4033   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   1stc1st 6782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785
This theorem is referenced by: (None)
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