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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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