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Theorem mpoxopoveq 6293
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
mpoxopoveq.f  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )
Assertion
Ref Expression
mpoxopoveq  |-  ( ( ( 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 mpoxopoveq
StepHypRef Expression
1 mpoxopoveq.f . . 3  |-  F  =  ( x  e.  _V ,  y  e.  ( 1st `  x )  |->  { n  e.  ( 1st `  x )  |  ph } )
21a1i 9 . 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 5554 . . . . 5  |-  ( x  =  <. V ,  W >.  ->  ( 1st `  x
)  =  ( 1st `  <. V ,  W >. ) )
4 op1stg 6203 . . . . . 6  |-  ( ( V  e.  X  /\  W  e.  Y )  ->  ( 1st `  <. V ,  W >. )  =  V )
54adantr 276 . . . . 5  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  ( 1st `  <. V ,  W >. )  =  V )
63, 5sylan9eqr 2248 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  x  =  <. V ,  W >. )  ->  ( 1st `  x )  =  V )
76adantrr 479 . . 3  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( 1st `  x
)  =  V )
8 sbceq1a 2995 . . . . . 6  |-  ( y  =  K  ->  ( ph 
<-> 
[. K  /  y ]. ph ) )
98adantl 277 . . . . 5  |-  ( ( x  =  <. V ,  W >.  /\  y  =  K )  ->  ( ph 
<-> 
[. K  /  y ]. ph ) )
109adantl 277 . . . 4  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( ph  <->  [. K  /  y ]. ph ) )
11 sbceq1a 2995 . . . . . 6  |-  ( x  =  <. V ,  W >.  ->  ( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph ) )
1211adantr 276 . . . . 5  |-  ( ( x  =  <. V ,  W >.  /\  y  =  K )  ->  ( [. K  /  y ]. ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
1312adantl 277 . . . 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 188 . . 3  |-  ( ( ( ( V  e.  X  /\  W  e.  Y )  /\  K  e.  V )  /\  (
x  =  <. V ,  W >.  /\  y  =  K ) )  -> 
( ph  <->  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph )
)
157, 14rabeqbidv 2755 . 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 opexg 4257 . . 3  |-  ( ( V  e.  X  /\  W  e.  Y )  -> 
<. V ,  W >.  e. 
_V )
1716adantr 276 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  <. V ,  W >.  e.  _V )
18 simpr 110 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  K  e.  V )
19 rabexg 4172 . . 3  |-  ( V  e.  X  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  e.  _V )
2019ad2antrr 488 . 2  |-  ( ( ( V  e.  X  /\  W  e.  Y
)  /\  K  e.  V )  ->  { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }  e.  _V )
21 equid 1712 . . 3  |-  z  =  z
22 nfvd 1540 . . 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 1540 . . 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 2336 . 2  |-  F/_ y <. V ,  W >.
27 nfcv 2336 . 2  |-  F/_ x K
28 nfsbc1v 3004 . . 3  |-  F/ x [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph
29 nfcv 2336 . . 3  |-  F/_ x V
3028, 29nfrabw 2675 . 2  |-  F/_ x { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
31 nfsbc1v 3004 . . . 4  |-  F/ y
[. K  /  y ]. ph
3226, 31nfsbc 3006 . . 3  |-  F/ y
[. <. V ,  W >.  /  x ]. [. K  /  y ]. ph
33 nfcv 2336 . . 3  |-  F/_ y V
3432, 33nfrabw 2675 . 2  |-  F/_ y { n  e.  V  |  [. <. V ,  W >.  /  x ]. [. K  /  y ]. ph }
352, 15, 6, 17, 18, 20, 23, 25, 26, 27, 30, 34ovmpodxf 6044 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    /\ wa 104    <-> wb 105    = wceq 1364   F/wnf 1471    e. wcel 2164   {crab 2476   _Vcvv 2760   [.wsbc 2985   <.cop 3621   ` cfv 5254  (class class class)co 5918    e. cmpo 5920   1stc1st 6191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193
This theorem is referenced by:  mpoxopovel  6294
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