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Theorem fimaproj 28660
Description: Image of a cartesian product for a function on pairs, given two projections  F and  G. (Contributed by Thierry Arnoux, 30-Dec-2019.)
Hypotheses
Ref Expression
fvproj.h  |-  H  =  ( x  e.  A ,  y  e.  B  |-> 
<. ( F `  x
) ,  ( G `
 y ) >.
)
fimaproj.f  |-  ( ph  ->  F  Fn  A )
fimaproj.g  |-  ( ph  ->  G  Fn  B )
fimaproj.x  |-  ( ph  ->  X  C_  A )
fimaproj.y  |-  ( ph  ->  Y  C_  B )
Assertion
Ref Expression
fimaproj  |-  ( ph  ->  ( H " ( X  X.  Y ) )  =  ( ( F
" X )  X.  ( G " Y
) ) )
Distinct variable groups:    x, A, y    x, B, y    x, F, y    x, G, y   
x, H, y
Allowed substitution hints:    ph( x, y)    X( x, y)    Y( x, y)

Proof of Theorem fimaproj
Dummy variables  a 
b  z  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4664 . . . . 5  |-  <. ( F `  ( 1st `  z ) ) ,  ( G `  ( 2nd `  z ) )
>.  e.  _V
2 fvproj.h . . . . . 6  |-  H  =  ( x  e.  A ,  y  e.  B  |-> 
<. ( F `  x
) ,  ( G `
 y ) >.
)
3 vex 3048 . . . . . . . . . 10  |-  x  e. 
_V
4 vex 3048 . . . . . . . . . 10  |-  y  e. 
_V
53, 4op1std 6803 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( 1st `  z
)  =  x )
65fveq2d 5869 . . . . . . . 8  |-  ( z  =  <. x ,  y
>.  ->  ( F `  ( 1st `  z ) )  =  ( F `
 x ) )
73, 4op2ndd 6804 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  y )
87fveq2d 5869 . . . . . . . 8  |-  ( z  =  <. x ,  y
>.  ->  ( G `  ( 2nd `  z ) )  =  ( G `
 y ) )
96, 8opeq12d 4174 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  <. ( F `  ( 1st `  z ) ) ,  ( G `
 ( 2nd `  z
) ) >.  =  <. ( F `  x ) ,  ( G `  y ) >. )
109mpt2mpt 6388 . . . . . 6  |-  ( z  e.  ( A  X.  B )  |->  <. ( F `  ( 1st `  z ) ) ,  ( G `  ( 2nd `  z ) )
>. )  =  (
x  e.  A , 
y  e.  B  |->  <.
( F `  x
) ,  ( G `
 y ) >.
)
112, 10eqtr4i 2476 . . . . 5  |-  H  =  ( z  e.  ( A  X.  B ) 
|->  <. ( F `  ( 1st `  z ) ) ,  ( G `
 ( 2nd `  z
) ) >. )
121, 11fnmpti 5706 . . . 4  |-  H  Fn  ( A  X.  B
)
13 fimaproj.x . . . . 5  |-  ( ph  ->  X  C_  A )
14 fimaproj.y . . . . 5  |-  ( ph  ->  Y  C_  B )
15 xpss12 4940 . . . . 5  |-  ( ( X  C_  A  /\  Y  C_  B )  -> 
( X  X.  Y
)  C_  ( A  X.  B ) )
1613, 14, 15syl2anc 667 . . . 4  |-  ( ph  ->  ( X  X.  Y
)  C_  ( A  X.  B ) )
17 fvelimab 5921 . . . 4  |-  ( ( H  Fn  ( A  X.  B )  /\  ( X  X.  Y
)  C_  ( A  X.  B ) )  -> 
( c  e.  ( H " ( X  X.  Y ) )  <->  E. z  e.  ( X  X.  Y ) ( H `  z )  =  c ) )
1812, 16, 17sylancr 669 . . 3  |-  ( ph  ->  ( c  e.  ( H " ( X  X.  Y ) )  <->  E. z  e.  ( X  X.  Y ) ( H `  z )  =  c ) )
19 simp-4r 777 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  a  e.  X )
20 simplr 762 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  b  e.  Y )
21 opelxpi 4866 . . . . . . . 8  |-  ( ( a  e.  X  /\  b  e.  Y )  -> 
<. a ,  b >.  e.  ( X  X.  Y
) )
2219, 20, 21syl2anc 667 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  <. a ,  b >.  e.  ( X  X.  Y ) )
23 simpllr 769 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  ( F `  a )  =  ( 1st `  c
) )
24 simpr 463 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  ( G `  b )  =  ( 2nd `  c
) )
2523, 24opeq12d 4174 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  <. ( F `  a ) ,  ( G `  b ) >.  =  <. ( 1st `  c ) ,  ( 2nd `  c
) >. )
2613ad5antr 740 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  X  C_  A )
2726, 19sseldd 3433 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  a  e.  A )
2814ad5antr 740 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  Y  C_  B )
2928, 20sseldd 3433 . . . . . . . . 9  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  b  e.  B )
302, 27, 29fvproj 28659 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  ( H `  <. a ,  b >. )  =  <. ( F `  a ) ,  ( G `  b ) >. )
31 1st2nd2 6830 . . . . . . . . 9  |-  ( c  e.  ( ( F
" X )  X.  ( G " Y
) )  ->  c  =  <. ( 1st `  c
) ,  ( 2nd `  c ) >. )
3231ad5antlr 741 . . . . . . . 8  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  c  =  <. ( 1st `  c
) ,  ( 2nd `  c ) >. )
3325, 30, 323eqtr4d 2495 . . . . . . 7  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  ( H `  <. a ,  b >. )  =  c )
34 fveq2 5865 . . . . . . . . 9  |-  ( z  =  <. a ,  b
>.  ->  ( H `  z )  =  ( H `  <. a ,  b >. )
)
3534eqeq1d 2453 . . . . . . . 8  |-  ( z  =  <. a ,  b
>.  ->  ( ( H `
 z )  =  c  <->  ( H `  <. a ,  b >.
)  =  c ) )
3635rspcev 3150 . . . . . . 7  |-  ( (
<. a ,  b >.  e.  ( X  X.  Y
)  /\  ( H `  <. a ,  b
>. )  =  c
)  ->  E. z  e.  ( X  X.  Y
) ( H `  z )  =  c )
3722, 33, 36syl2anc 667 . . . . . 6  |-  ( ( ( ( ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  /\  a  e.  X )  /\  ( F `  a )  =  ( 1st `  c
) )  /\  b  e.  Y )  /\  ( G `  b )  =  ( 2nd `  c
) )  ->  E. z  e.  ( X  X.  Y
) ( H `  z )  =  c )
38 fimaproj.g . . . . . . . . 9  |-  ( ph  ->  G  Fn  B )
3938ad3antrrr 736 . . . . . . . 8  |-  ( ( ( ( ph  /\  c  e.  ( ( F " X )  X.  ( G " Y
) ) )  /\  a  e.  X )  /\  ( F `  a
)  =  ( 1st `  c ) )  ->  G  Fn  B )
40 fnfun 5673 . . . . . . . 8  |-  ( G  Fn  B  ->  Fun  G )
4139, 40syl 17 . . . . . . 7  |-  ( ( ( ( ph  /\  c  e.  ( ( F " X )  X.  ( G " Y
) ) )  /\  a  e.  X )  /\  ( F `  a
)  =  ( 1st `  c ) )  ->  Fun  G )
42 xp2nd 6824 . . . . . . . 8  |-  ( c  e.  ( ( F
" X )  X.  ( G " Y
) )  ->  ( 2nd `  c )  e.  ( G " Y
) )
4342ad3antlr 737 . . . . . . 7  |-  ( ( ( ( ph  /\  c  e.  ( ( F " X )  X.  ( G " Y
) ) )  /\  a  e.  X )  /\  ( F `  a
)  =  ( 1st `  c ) )  -> 
( 2nd `  c
)  e.  ( G
" Y ) )
44 fvelima 5917 . . . . . . 7  |-  ( ( Fun  G  /\  ( 2nd `  c )  e.  ( G " Y
) )  ->  E. b  e.  Y  ( G `  b )  =  ( 2nd `  c ) )
4541, 43, 44syl2anc 667 . . . . . 6  |-  ( ( ( ( ph  /\  c  e.  ( ( F " X )  X.  ( G " Y
) ) )  /\  a  e.  X )  /\  ( F `  a
)  =  ( 1st `  c ) )  ->  E. b  e.  Y  ( G `  b )  =  ( 2nd `  c
) )
4637, 45r19.29a 2932 . . . . 5  |-  ( ( ( ( ph  /\  c  e.  ( ( F " X )  X.  ( G " Y
) ) )  /\  a  e.  X )  /\  ( F `  a
)  =  ( 1st `  c ) )  ->  E. z  e.  ( X  X.  Y ) ( H `  z )  =  c )
47 fimaproj.f . . . . . . . 8  |-  ( ph  ->  F  Fn  A )
4847adantr 467 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  ->  F  Fn  A )
49 fnfun 5673 . . . . . . 7  |-  ( F  Fn  A  ->  Fun  F )
5048, 49syl 17 . . . . . 6  |-  ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  ->  Fun  F )
51 xp1st 6823 . . . . . . 7  |-  ( c  e.  ( ( F
" X )  X.  ( G " Y
) )  ->  ( 1st `  c )  e.  ( F " X
) )
5251adantl 468 . . . . . 6  |-  ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  ->  ( 1st `  c )  e.  ( F " X ) )
53 fvelima 5917 . . . . . 6  |-  ( ( Fun  F  /\  ( 1st `  c )  e.  ( F " X
) )  ->  E. a  e.  X  ( F `  a )  =  ( 1st `  c ) )
5450, 52, 53syl2anc 667 . . . . 5  |-  ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  ->  E. a  e.  X  ( F `  a )  =  ( 1st `  c ) )
5546, 54r19.29a 2932 . . . 4  |-  ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  ->  E. z  e.  ( X  X.  Y
) ( H `  z )  =  c )
56 simpr 463 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( H `  z
)  =  c )
5716ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( X  X.  Y
)  C_  ( A  X.  B ) )
58 simplr 762 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
z  e.  ( X  X.  Y ) )
5957, 58sseldd 3433 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
z  e.  ( A  X.  B ) )
6011fvmpt2 5957 . . . . . . . 8  |-  ( ( z  e.  ( A  X.  B )  /\  <.
( F `  ( 1st `  z ) ) ,  ( G `  ( 2nd `  z ) ) >.  e.  _V )  ->  ( H `  z )  =  <. ( F `  ( 1st `  z ) ) ,  ( G `  ( 2nd `  z ) )
>. )
6159, 1, 60sylancl 668 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( H `  z
)  =  <. ( F `  ( 1st `  z ) ) ,  ( G `  ( 2nd `  z ) )
>. )
6247ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  ->  F  Fn  A )
6313ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  ->  X  C_  A )
64 xp1st 6823 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 1st `  z )  e.  X )
6558, 64syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( 1st `  z
)  e.  X )
66 fnfvima 6143 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  X  C_  A  /\  ( 1st `  z )  e.  X )  ->  ( F `  ( 1st `  z ) )  e.  ( F " X
) )
6762, 63, 65, 66syl3anc 1268 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( F `  ( 1st `  z ) )  e.  ( F " X ) )
6838ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  ->  G  Fn  B )
6914ad2antrr 732 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  ->  Y  C_  B )
70 xp2nd 6824 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 2nd `  z )  e.  Y )
7158, 70syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( 2nd `  z
)  e.  Y )
72 fnfvima 6143 . . . . . . . . 9  |-  ( ( G  Fn  B  /\  Y  C_  B  /\  ( 2nd `  z )  e.  Y )  ->  ( G `  ( 2nd `  z ) )  e.  ( G " Y
) )
7368, 69, 71, 72syl3anc 1268 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( G `  ( 2nd `  z ) )  e.  ( G " Y ) )
74 opelxpi 4866 . . . . . . . 8  |-  ( ( ( F `  ( 1st `  z ) )  e.  ( F " X )  /\  ( G `  ( 2nd `  z ) )  e.  ( G " Y
) )  ->  <. ( F `  ( 1st `  z ) ) ,  ( G `  ( 2nd `  z ) )
>.  e.  ( ( F
" X )  X.  ( G " Y
) ) )
7567, 73, 74syl2anc 667 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  ->  <. ( F `  ( 1st `  z ) ) ,  ( G `  ( 2nd `  z ) ) >.  e.  (
( F " X
)  X.  ( G
" Y ) ) )
7661, 75eqeltrd 2529 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( H `  z
)  e.  ( ( F " X )  X.  ( G " Y ) ) )
7756, 76eqeltrrd 2530 . . . . 5  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
c  e.  ( ( F " X )  X.  ( G " Y ) ) )
7877r19.29an 2931 . . . 4  |-  ( (
ph  /\  E. z  e.  ( X  X.  Y
) ( H `  z )  =  c )  ->  c  e.  ( ( F " X )  X.  ( G " Y ) ) )
7955, 78impbida 843 . . 3  |-  ( ph  ->  ( c  e.  ( ( F " X
)  X.  ( G
" Y ) )  <->  E. z  e.  ( X  X.  Y ) ( H `  z )  =  c ) )
8018, 79bitr4d 260 . 2  |-  ( ph  ->  ( c  e.  ( H " ( X  X.  Y ) )  <-> 
c  e.  ( ( F " X )  X.  ( G " Y ) ) ) )
8180eqrdv 2449 1  |-  ( ph  ->  ( H " ( X  X.  Y ) )  =  ( ( F
" X )  X.  ( G " Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   E.wrex 2738   _Vcvv 3045    C_ wss 3404   <.cop 3974    |-> cmpt 4461    X. cxp 4832   "cima 4837   Fun wfun 5576    Fn wfn 5577   ` cfv 5582    |-> cmpt2 6292   1stc1st 6791   2ndc2nd 6792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794
This theorem is referenced by:  txomap  28661
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