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Theorem mpt2xopoveq 6950
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 11 . 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 5849 . . . . 5  |-  ( x  =  <. V ,  W >.  ->  ( 1st `  x
)  =  ( 1st `  <. V ,  W >. ) )
4 op1stg 6796 . . . . . 6  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( 1st `  <. V ,  W >. )  =  V )
54adantr 463 . . . . 5  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( 1st `  <. V ,  W >. )  =  V )
63, 5sylan9eqr 2465 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  x  =  <. V ,  W >. )  ->  ( 1st `  x )  =  V )
76adantrr 715 . . 3  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( 1st `  x
)  =  V )
8 sbceq1a 3288 . . . . . 6  |-  ( y  =  K  ->  ( ph 
<-> 
[. K  /  y ]. ph ) )
98adantl 464 . . . . 5  |-  ( ( x  =  <. V ,  W >.  /\  y  =  K )  ->  ( ph 
<-> 
[. K  /  y ]. ph ) )
109adantl 464 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( ph  <->  [. K  /  y ]. ph ) )
11 sbceq1a 3288 . . . . . 6  |-  ( x  =  <. V ,  W >.  ->  ( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph ) )
1211adantr 463 . . . . 5  |-  ( ( x  =  <. V ,  W >.  /\  y  =  K )  ->  ( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
1312adantl 464 . . . 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 253 . . 3  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
157, 14rabeqbidv 3054 . 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 opex 4655 . . 3  |-  <. V ,  W >.  e.  _V
1716a1i 11 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  <. V ,  W >.  e.  _V )
18 simpr 459 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  K  e.  V )
19 rabexg 4544 . . 3  |-  ( V  e.  X  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  e.  _V )
2019ad2antrr 724 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  e.  _V )
21 equid 1815 . . 3  |-  z  =  z
22 nfvd 1729 . . 3  |-  ( z  =  z  ->  F/ x ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V ) )
2321, 22ax-mp 5 . 2  |-  F/ x
( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )
24 nfvd 1729 . . 3  |-  ( z  =  z  ->  F/ y ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V ) )
2521, 24ax-mp 5 . 2  |-  F/ y ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )
26 nfcv 2564 . 2  |-  F/_ y <. V ,  W >.
27 nfcv 2564 . 2  |-  F/_ x K
28 nfsbc1v 3297 . . 3  |-  F/ x [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph
29 nfcv 2564 . . 3  |-  F/_ x V
3028, 29nfrab 2989 . 2  |-  F/_ x { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
31 nfsbc1v 3297 . . . 4  |-  F/ y
[. K  /  y ]. ph
3226, 31nfsbc 3299 . . 3  |-  F/ y
[. <. V ,  W >.  /  x ]. [. K  /  y ]. ph
33 nfcv 2564 . . 3  |-  F/_ y V
3432, 33nfrab 2989 . 2  |-  F/_ y { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpt2dxf 6409 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   F/wnf 1637    e. wcel 1842   {crab 2758   _Vcvv 3059   [.wsbc 3277   <.cop 3978   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   1stc1st 6782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784
This theorem is referenced by:  mpt2xopovel  6951  mpt2xopoveqd  6952  nbgraopALT  24841
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