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Theorem fimaproj 28084
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 4720 . . . . 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 3112 . . . . . . . . . 10  |-  x  e. 
_V
4 vex 3112 . . . . . . . . . 10  |-  y  e. 
_V
53, 4op1std 6809 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( 1st `  z
)  =  x )
65fveq2d 5876 . . . . . . . 8  |-  ( z  =  <. x ,  y
>.  ->  ( F `  ( 1st `  z ) )  =  ( F `
 x ) )
73, 4op2ndd 6810 . . . . . . . . 9  |-  ( z  =  <. x ,  y
>.  ->  ( 2nd `  z
)  =  y )
87fveq2d 5876 . . . . . . . 8  |-  ( z  =  <. x ,  y
>.  ->  ( G `  ( 2nd `  z ) )  =  ( G `
 y ) )
96, 8opeq12d 4227 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  <. ( F `  ( 1st `  z ) ) ,  ( G `
 ( 2nd `  z
) ) >.  =  <. ( F `  x ) ,  ( G `  y ) >. )
109mpt2mpt 6393 . . . . . 6  |-  ( z  e.  ( A  X.  B )  |->  <. ( F `  ( 1st `  z ) ) ,  ( G `  ( 2nd `  z ) )
>. )  =  (
x  e.  A , 
y  e.  B  |->  <.
( F `  x
) ,  ( G `
 y ) >.
)
112, 10eqtr4i 2489 . . . . 5  |-  H  =  ( z  e.  ( A  X.  B ) 
|->  <. ( F `  ( 1st `  z ) ) ,  ( G `
 ( 2nd `  z
) ) >. )
121, 11fnmpti 5715 . . . 4  |-  H  Fn  ( A  X.  B
)
13 fimaproj.x . . . . 5  |-  ( ph  ->  X  C_  A )
14 fimaproj.y . . . . 5  |-  ( ph  ->  Y  C_  B )
15 xpss12 5117 . . . . 5  |-  ( ( X  C_  A  /\  Y  C_  B )  -> 
( X  X.  Y
)  C_  ( A  X.  B ) )
1613, 14, 15syl2anc 661 . . . 4  |-  ( ph  ->  ( X  X.  Y
)  C_  ( A  X.  B ) )
17 fvelimab 5929 . . . 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 663 . . 3  |-  ( ph  ->  ( c  e.  ( H " ( X  X.  Y ) )  <->  E. z  e.  ( X  X.  Y ) ( H `  z )  =  c ) )
19 simp-4r 768 . . . . . . . 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 755 . . . . . . . 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 5040 . . . . . . . 8  |-  ( ( a  e.  X  /\  b  e.  Y )  -> 
<. a ,  b >.  e.  ( X  X.  Y
) )
2219, 20, 21syl2anc 661 . . . . . . 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 760 . . . . . . . . 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 461 . . . . . . . . 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 4227 . . . . . . . 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 733 . . . . . . . . . 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 3500 . . . . . . . . 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 733 . . . . . . . . . 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 3500 . . . . . . . . 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 28083 . . . . . . . 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 6836 . . . . . . . . 9  |-  ( c  e.  ( ( F
" X )  X.  ( G " Y
) )  ->  c  =  <. ( 1st `  c
) ,  ( 2nd `  c ) >. )
3231ad5antlr 734 . . . . . . . 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 2508 . . . . . . 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 5872 . . . . . . . . 9  |-  ( z  =  <. a ,  b
>.  ->  ( H `  z )  =  ( H `  <. a ,  b >. )
)
3534eqeq1d 2459 . . . . . . . 8  |-  ( z  =  <. a ,  b
>.  ->  ( ( H `
 z )  =  c  <->  ( H `  <. a ,  b >.
)  =  c ) )
3635rspcev 3210 . . . . . . 7  |-  ( (
<. a ,  b >.  e.  ( X  X.  Y
)  /\  ( H `  <. a ,  b
>. )  =  c
)  ->  E. z  e.  ( X  X.  Y
) ( H `  z )  =  c )
3722, 33, 36syl2anc 661 . . . . . 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 729 . . . . . . . 8  |-  ( ( ( ( ph  /\  c  e.  ( ( F " X )  X.  ( G " Y
) ) )  /\  a  e.  X )  /\  ( F `  a
)  =  ( 1st `  c ) )  ->  G  Fn  B )
40 fnfun 5684 . . . . . . . 8  |-  ( G  Fn  B  ->  Fun  G )
4139, 40syl 16 . . . . . . 7  |-  ( ( ( ( ph  /\  c  e.  ( ( F " X )  X.  ( G " Y
) ) )  /\  a  e.  X )  /\  ( F `  a
)  =  ( 1st `  c ) )  ->  Fun  G )
42 xp2nd 6830 . . . . . . . 8  |-  ( c  e.  ( ( F
" X )  X.  ( G " Y
) )  ->  ( 2nd `  c )  e.  ( G " Y
) )
4342ad3antlr 730 . . . . . . 7  |-  ( ( ( ( ph  /\  c  e.  ( ( F " X )  X.  ( G " Y
) ) )  /\  a  e.  X )  /\  ( F `  a
)  =  ( 1st `  c ) )  -> 
( 2nd `  c
)  e.  ( G
" Y ) )
44 fvelima 5925 . . . . . . 7  |-  ( ( Fun  G  /\  ( 2nd `  c )  e.  ( G " Y
) )  ->  E. b  e.  Y  ( G `  b )  =  ( 2nd `  c ) )
4541, 43, 44syl2anc 661 . . . . . 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 2999 . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  ->  F  Fn  A )
49 fnfun 5684 . . . . . . 7  |-  ( F  Fn  A  ->  Fun  F )
5048, 49syl 16 . . . . . 6  |-  ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  ->  Fun  F )
51 xp1st 6829 . . . . . . 7  |-  ( c  e.  ( ( F
" X )  X.  ( G " Y
) )  ->  ( 1st `  c )  e.  ( F " X
) )
5251adantl 466 . . . . . 6  |-  ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  ->  ( 1st `  c )  e.  ( F " X ) )
53 fvelima 5925 . . . . . 6  |-  ( ( Fun  F  /\  ( 1st `  c )  e.  ( F " X
) )  ->  E. a  e.  X  ( F `  a )  =  ( 1st `  c ) )
5450, 52, 53syl2anc 661 . . . . 5  |-  ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  ->  E. a  e.  X  ( F `  a )  =  ( 1st `  c ) )
5546, 54r19.29a 2999 . . . 4  |-  ( (
ph  /\  c  e.  ( ( F " X )  X.  ( G " Y ) ) )  ->  E. z  e.  ( X  X.  Y
) ( H `  z )  =  c )
56 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( H `  z
)  =  c )
5716ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( X  X.  Y
)  C_  ( A  X.  B ) )
58 simplr 755 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
z  e.  ( X  X.  Y ) )
5957, 58sseldd 3500 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
z  e.  ( A  X.  B ) )
6011fvmpt2 5964 . . . . . . . 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 662 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( H `  z
)  =  <. ( F `  ( 1st `  z ) ) ,  ( G `  ( 2nd `  z ) )
>. )
6247ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  ->  F  Fn  A )
6313ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  ->  X  C_  A )
64 xp1st 6829 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 1st `  z )  e.  X )
6558, 64syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( 1st `  z
)  e.  X )
66 fnfvima 6151 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  X  C_  A  /\  ( 1st `  z )  e.  X )  ->  ( F `  ( 1st `  z ) )  e.  ( F " X
) )
6762, 63, 65, 66syl3anc 1228 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( F `  ( 1st `  z ) )  e.  ( F " X ) )
6838ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  ->  G  Fn  B )
6914ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  ->  Y  C_  B )
70 xp2nd 6830 . . . . . . . . . 10  |-  ( z  e.  ( X  X.  Y )  ->  ( 2nd `  z )  e.  Y )
7158, 70syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( 2nd `  z
)  e.  Y )
72 fnfvima 6151 . . . . . . . . 9  |-  ( ( G  Fn  B  /\  Y  C_  B  /\  ( 2nd `  z )  e.  Y )  ->  ( G `  ( 2nd `  z ) )  e.  ( G " Y
) )
7368, 69, 71, 72syl3anc 1228 . . . . . . . 8  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( G `  ( 2nd `  z ) )  e.  ( G " Y ) )
74 opelxpi 5040 . . . . . . . 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 661 . . . . . . 7  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  ->  <. ( F `  ( 1st `  z ) ) ,  ( G `  ( 2nd `  z ) ) >.  e.  (
( F " X
)  X.  ( G
" Y ) ) )
7661, 75eqeltrd 2545 . . . . . 6  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
( H `  z
)  e.  ( ( F " X )  X.  ( G " Y ) ) )
7756, 76eqeltrrd 2546 . . . . 5  |-  ( ( ( ph  /\  z  e.  ( X  X.  Y
) )  /\  ( H `  z )  =  c )  -> 
c  e.  ( ( F " X )  X.  ( G " Y ) ) )
7877r19.29an 2998 . . . 4  |-  ( (
ph  /\  E. z  e.  ( X  X.  Y
) ( H `  z )  =  c )  ->  c  e.  ( ( F " X )  X.  ( G " Y ) ) )
7955, 78impbida 832 . . 3  |-  ( ph  ->  ( c  e.  ( ( F " X
)  X.  ( G
" Y ) )  <->  E. z  e.  ( X  X.  Y ) ( H `  z )  =  c ) )
8018, 79bitr4d 256 . 2  |-  ( ph  ->  ( c  e.  ( H " ( X  X.  Y ) )  <-> 
c  e.  ( ( F " X )  X.  ( G " Y ) ) ) )
8180eqrdv 2454 1  |-  ( ph  ->  ( H " ( X  X.  Y ) )  =  ( ( F
" X )  X.  ( G " Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109    C_ wss 3471   <.cop 4038    |-> cmpt 4515    X. cxp 5006   "cima 5011   Fun wfun 5588    Fn wfn 5589   ` cfv 5594    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800
This theorem is referenced by:  txomap  28085
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