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Theorem mpt2eq123 5907
Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) (Revised by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
mpt2eq123  |-  ( ( A  =  D  /\  A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F ) )  -> 
( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
Distinct variable groups:    x, y, A    y, B    x, D, y    y, E
Allowed substitution hints:    B( x)    C( x, y)    E( x)    F( x, y)

Proof of Theorem mpt2eq123
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1605 . . . 4  |-  F/ x  A  =  D
2 nfra1 2593 . . . 4  |-  F/ x A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F )
31, 2nfan 1771 . . 3  |-  F/ x
( A  =  D  /\  A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F ) )
4 nfv 1605 . . . 4  |-  F/ y  A  =  D
5 nfcv 2419 . . . . 5  |-  F/_ y A
6 nfv 1605 . . . . . 6  |-  F/ y  B  =  E
7 nfra1 2593 . . . . . 6  |-  F/ y A. y  e.  B  C  =  F
86, 7nfan 1771 . . . . 5  |-  F/ y ( B  =  E  /\  A. y  e.  B  C  =  F )
95, 8nfral 2596 . . . 4  |-  F/ y A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F )
104, 9nfan 1771 . . 3  |-  F/ y ( A  =  D  /\  A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F ) )
11 nfv 1605 . . 3  |-  F/ z ( A  =  D  /\  A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F ) )
12 rsp 2603 . . . . . . 7  |-  ( A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F )  ->  (
x  e.  A  -> 
( B  =  E  /\  A. y  e.  B  C  =  F ) ) )
13 rsp 2603 . . . . . . . . . 10  |-  ( A. y  e.  B  C  =  F  ->  ( y  e.  B  ->  C  =  F ) )
14 eqeq2 2292 . . . . . . . . . 10  |-  ( C  =  F  ->  (
z  =  C  <->  z  =  F ) )
1513, 14syl6 29 . . . . . . . . 9  |-  ( A. y  e.  B  C  =  F  ->  ( y  e.  B  ->  (
z  =  C  <->  z  =  F ) ) )
1615pm5.32d 620 . . . . . . . 8  |-  ( A. y  e.  B  C  =  F  ->  ( ( y  e.  B  /\  z  =  C )  <->  ( y  e.  B  /\  z  =  F )
) )
17 eleq2 2344 . . . . . . . . 9  |-  ( B  =  E  ->  (
y  e.  B  <->  y  e.  E ) )
1817anbi1d 685 . . . . . . . 8  |-  ( B  =  E  ->  (
( y  e.  B  /\  z  =  F
)  <->  ( y  e.  E  /\  z  =  F ) ) )
1916, 18sylan9bbr 681 . . . . . . 7  |-  ( ( B  =  E  /\  A. y  e.  B  C  =  F )  ->  (
( y  e.  B  /\  z  =  C
)  <->  ( y  e.  E  /\  z  =  F ) ) )
2012, 19syl6 29 . . . . . 6  |-  ( A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F )  ->  (
x  e.  A  -> 
( ( y  e.  B  /\  z  =  C )  <->  ( y  e.  E  /\  z  =  F ) ) ) )
2120pm5.32d 620 . . . . 5  |-  ( A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F )  ->  (
( x  e.  A  /\  ( y  e.  B  /\  z  =  C
) )  <->  ( x  e.  A  /\  (
y  e.  E  /\  z  =  F )
) ) )
22 eleq2 2344 . . . . . 6  |-  ( A  =  D  ->  (
x  e.  A  <->  x  e.  D ) )
2322anbi1d 685 . . . . 5  |-  ( A  =  D  ->  (
( x  e.  A  /\  ( y  e.  E  /\  z  =  F
) )  <->  ( x  e.  D  /\  (
y  e.  E  /\  z  =  F )
) ) )
2421, 23sylan9bbr 681 . . . 4  |-  ( ( A  =  D  /\  A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F ) )  -> 
( ( x  e.  A  /\  ( y  e.  B  /\  z  =  C ) )  <->  ( x  e.  D  /\  (
y  e.  E  /\  z  =  F )
) ) )
25 anass 630 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  z  =  C )  <->  ( x  e.  A  /\  (
y  e.  B  /\  z  =  C )
) )
26 anass 630 . . . 4  |-  ( ( ( x  e.  D  /\  y  e.  E
)  /\  z  =  F )  <->  ( x  e.  D  /\  (
y  e.  E  /\  z  =  F )
) )
2724, 25, 263bitr4g 279 . . 3  |-  ( ( A  =  D  /\  A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F ) )  -> 
( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) ) )
283, 10, 11, 27oprabbid 5901 . 2  |-  ( ( A  =  D  /\  A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F ) )  ->  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C ) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  D  /\  y  e.  E )  /\  z  =  F ) } )
29 df-mpt2 5863 . 2  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  C
) }
30 df-mpt2 5863 . 2  |-  ( x  e.  D ,  y  e.  E  |->  F )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  D  /\  y  e.  E )  /\  z  =  F
) }
3128, 29, 303eqtr4g 2340 1  |-  ( ( A  =  D  /\  A. x  e.  A  ( B  =  E  /\  A. y  e.  B  C  =  F ) )  -> 
( x  e.  A ,  y  e.  B  |->  C )  =  ( x  e.  D , 
y  e.  E  |->  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {coprab 5859    e. cmpt2 5860
This theorem is referenced by:  mpt2eq12  5908  mapxpen  7027  xkoptsub  17348  xkocnv  17505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-oprab 5862  df-mpt2 5863
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