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Theorem mpt2xopoveq 6473
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 5731 . . . . 5  |-  ( x  =  <. V ,  W >.  ->  ( 1st `  x
)  =  ( 1st `  <. V ,  W >. ) )
4 op1stg 6362 . . . . . 6  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( 1st `  <. V ,  W >. )  =  V )
54adantr 453 . . . . 5  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( 1st `  <. V ,  W >. )  =  V )
63, 5sylan9eqr 2492 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  x  =  <. V ,  W >. )  ->  ( 1st `  x )  =  V )
76adantrr 699 . . 3  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( 1st `  x
)  =  V )
8 sbceq1a 3173 . . . . . 6  |-  ( y  =  K  ->  ( ph 
<-> 
[. K  /  y ]. ph ) )
98adantl 454 . . . . 5  |-  ( ( x  =  <. V ,  W >.  /\  y  =  K )  ->  ( ph 
<-> 
[. K  /  y ]. ph ) )
109adantl 454 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( ph  <->  [. K  /  y ]. ph ) )
11 sbceq1a 3173 . . . . . 6  |-  ( x  =  <. V ,  W >.  ->  ( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph ) )
1211adantr 453 . . . . 5  |-  ( ( x  =  <. V ,  W >.  /\  y  =  K )  ->  ( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
1312adantl 454 . . . 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 246 . . 3  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
157, 14rabeqbidv 2953 . 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 4430 . . 3  |-  <. V ,  W >.  e.  _V
1716a1i 11 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  <. V ,  W >.  e.  _V )
18 simpr 449 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  K  e.  V )
19 rabexg 4356 . . 3  |-  ( V  e.  X  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  e.  _V )
2019ad2antrr 708 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  e.  _V )
21 equid 1689 . . 3  |-  z  =  z
22 nfvd 1631 . . 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 1631 . . 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 2574 . 2  |-  F/_ y <. V ,  W >.
27 nfcv 2574 . 2  |-  F/_ x K
28 nfsbc1v 3182 . . 3  |-  F/ x [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph
29 nfcv 2574 . . 3  |-  F/_ x V
3028, 29nfrab 2891 . 2  |-  F/_ x { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
31 nfsbc1v 3182 . . . 4  |-  F/ y
[. K  /  y ]. ph
3226, 31nfsbc 3184 . . 3  |-  F/ y
[. <. V ,  W >.  /  x ]. [. K  /  y ]. ph
33 nfcv 2574 . . 3  |-  F/_ y V
3432, 33nfrab 2891 . 2  |-  F/_ y { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpt2dxf 6202 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    <-> wb 178    /\ wa 360   F/wnf 1554    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958   [.wsbc 3163   <.cop 3819   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350
This theorem is referenced by:  mpt2xopovel  6474  mpt2xopoveqd  6475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352
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