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Theorem xppreima2 27709
Description: The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)
Hypotheses
Ref Expression
xppreima2.1  |-  ( ph  ->  F : A --> B )
xppreima2.2  |-  ( ph  ->  G : A --> C )
xppreima2.3  |-  H  =  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
)
Assertion
Ref Expression
xppreima2  |-  ( ph  ->  ( `' H "
( Y  X.  Z
) )  =  ( ( `' F " Y )  i^i  ( `' G " Z ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, F    x, G    x, H    ph, x
Allowed substitution hints:    Y( x)    Z( x)

Proof of Theorem xppreima2
StepHypRef Expression
1 xppreima2.3 . . . 4  |-  H  =  ( x  e.  A  |-> 
<. ( F `  x
) ,  ( G `
 x ) >.
)
21funmpt2 5607 . . 3  |-  Fun  H
3 xppreima2.1 . . . . . . . 8  |-  ( ph  ->  F : A --> B )
43ffvelrnda 6007 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( F `  x )  e.  B )
5 xppreima2.2 . . . . . . . 8  |-  ( ph  ->  G : A --> C )
65ffvelrnda 6007 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  ( G `  x )  e.  C )
7 opelxp 5018 . . . . . . 7  |-  ( <.
( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
)  <->  ( ( F `
 x )  e.  B  /\  ( G `
 x )  e.  C ) )
84, 6, 7sylanbrc 662 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  <. ( F `  x ) ,  ( G `  x ) >.  e.  ( B  X.  C ) )
98, 1fmptd 6031 . . . . 5  |-  ( ph  ->  H : A --> ( B  X.  C ) )
10 frn 5719 . . . . 5  |-  ( H : A --> ( B  X.  C )  ->  ran  H  C_  ( B  X.  C ) )
119, 10syl 16 . . . 4  |-  ( ph  ->  ran  H  C_  ( B  X.  C ) )
12 xpss 5097 . . . 4  |-  ( B  X.  C )  C_  ( _V  X.  _V )
1311, 12syl6ss 3501 . . 3  |-  ( ph  ->  ran  H  C_  ( _V  X.  _V ) )
14 xppreima 27708 . . 3  |-  ( ( Fun  H  /\  ran  H 
C_  ( _V  X.  _V ) )  ->  ( `' H " ( Y  X.  Z ) )  =  ( ( `' ( 1st  o.  H
) " Y )  i^i  ( `' ( 2nd  o.  H )
" Z ) ) )
152, 13, 14sylancr 661 . 2  |-  ( ph  ->  ( `' H "
( Y  X.  Z
) )  =  ( ( `' ( 1st 
o.  H ) " Y )  i^i  ( `' ( 2nd  o.  H ) " Z
) ) )
16 fo1st 6793 . . . . . . . . 9  |-  1st : _V -onto-> _V
17 fofn 5779 . . . . . . . . 9  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
1816, 17ax-mp 5 . . . . . . . 8  |-  1st  Fn  _V
19 opex 4701 . . . . . . . . 9  |-  <. ( F `  x ) ,  ( G `  x ) >.  e.  _V
2019, 1fnmpti 5691 . . . . . . . 8  |-  H  Fn  A
21 ssv 3509 . . . . . . . 8  |-  ran  H  C_ 
_V
22 fnco 5671 . . . . . . . 8  |-  ( ( 1st  Fn  _V  /\  H  Fn  A  /\  ran  H  C_  _V )  ->  ( 1st  o.  H
)  Fn  A )
2318, 20, 21, 22mp3an 1322 . . . . . . 7  |-  ( 1st 
o.  H )  Fn  A
2423a1i 11 . . . . . 6  |-  ( ph  ->  ( 1st  o.  H
)  Fn  A )
25 ffn 5713 . . . . . . 7  |-  ( F : A --> B  ->  F  Fn  A )
263, 25syl 16 . . . . . 6  |-  ( ph  ->  F  Fn  A )
272a1i 11 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  Fun  H )
2813adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  ran  H 
C_  ( _V  X.  _V ) )
29 simpr 459 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  A )
3019, 1dmmpti 5692 . . . . . . . . . . 11  |-  dom  H  =  A
3129, 30syl6eleqr 2553 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  dom  H )
32 opfv 27707 . . . . . . . . . 10  |-  ( ( ( Fun  H  /\  ran  H  C_  ( _V  X.  _V ) )  /\  x  e.  dom  H )  ->  ( H `  x )  =  <. ( ( 1st  o.  H
) `  x ) ,  ( ( 2nd 
o.  H ) `  x ) >. )
3327, 28, 31, 32syl21anc 1225 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( H `  x )  =  <. ( ( 1st 
o.  H ) `  x ) ,  ( ( 2nd  o.  H
) `  x ) >. )
341fvmpt2 5939 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  <.
( F `  x
) ,  ( G `
 x ) >.  e.  ( B  X.  C
) )  ->  ( H `  x )  =  <. ( F `  x ) ,  ( G `  x )
>. )
3529, 8, 34syl2anc 659 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  ( H `  x )  =  <. ( F `  x ) ,  ( G `  x )
>. )
3633, 35eqtr3d 2497 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  <. (
( 1st  o.  H
) `  x ) ,  ( ( 2nd 
o.  H ) `  x ) >.  =  <. ( F `  x ) ,  ( G `  x ) >. )
37 fvex 5858 . . . . . . . . 9  |-  ( ( 1st  o.  H ) `
 x )  e. 
_V
38 fvex 5858 . . . . . . . . 9  |-  ( ( 2nd  o.  H ) `
 x )  e. 
_V
3937, 38opth 4711 . . . . . . . 8  |-  ( <.
( ( 1st  o.  H ) `  x
) ,  ( ( 2nd  o.  H ) `
 x ) >.  =  <. ( F `  x ) ,  ( G `  x )
>. 
<->  ( ( ( 1st 
o.  H ) `  x )  =  ( F `  x )  /\  ( ( 2nd 
o.  H ) `  x )  =  ( G `  x ) ) )
4036, 39sylib 196 . . . . . . 7  |-  ( (
ph  /\  x  e.  A )  ->  (
( ( 1st  o.  H ) `  x
)  =  ( F `
 x )  /\  ( ( 2nd  o.  H ) `  x
)  =  ( G `
 x ) ) )
4140simpld 457 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( 1st  o.  H
) `  x )  =  ( F `  x ) )
4224, 26, 41eqfnfvd 5960 . . . . 5  |-  ( ph  ->  ( 1st  o.  H
)  =  F )
4342cnveqd 5167 . . . 4  |-  ( ph  ->  `' ( 1st  o.  H )  =  `' F )
4443imaeq1d 5324 . . 3  |-  ( ph  ->  ( `' ( 1st 
o.  H ) " Y )  =  ( `' F " Y ) )
45 fo2nd 6794 . . . . . . . . 9  |-  2nd : _V -onto-> _V
46 fofn 5779 . . . . . . . . 9  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
4745, 46ax-mp 5 . . . . . . . 8  |-  2nd  Fn  _V
48 fnco 5671 . . . . . . . 8  |-  ( ( 2nd  Fn  _V  /\  H  Fn  A  /\  ran  H  C_  _V )  ->  ( 2nd  o.  H
)  Fn  A )
4947, 20, 21, 48mp3an 1322 . . . . . . 7  |-  ( 2nd 
o.  H )  Fn  A
5049a1i 11 . . . . . 6  |-  ( ph  ->  ( 2nd  o.  H
)  Fn  A )
51 ffn 5713 . . . . . . 7  |-  ( G : A --> C  ->  G  Fn  A )
525, 51syl 16 . . . . . 6  |-  ( ph  ->  G  Fn  A )
5340simprd 461 . . . . . 6  |-  ( (
ph  /\  x  e.  A )  ->  (
( 2nd  o.  H
) `  x )  =  ( G `  x ) )
5450, 52, 53eqfnfvd 5960 . . . . 5  |-  ( ph  ->  ( 2nd  o.  H
)  =  G )
5554cnveqd 5167 . . . 4  |-  ( ph  ->  `' ( 2nd  o.  H )  =  `' G )
5655imaeq1d 5324 . . 3  |-  ( ph  ->  ( `' ( 2nd 
o.  H ) " Z )  =  ( `' G " Z ) )
5744, 56ineq12d 3687 . 2  |-  ( ph  ->  ( ( `' ( 1st  o.  H )
" Y )  i^i  ( `' ( 2nd 
o.  H ) " Z ) )  =  ( ( `' F " Y )  i^i  ( `' G " Z ) ) )
5815, 57eqtrd 2495 1  |-  ( ph  ->  ( `' H "
( Y  X.  Z
) )  =  ( ( `' F " Y )  i^i  ( `' G " Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    i^i cin 3460    C_ wss 3461   <.cop 4022    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991    o. ccom 4992   Fun wfun 5564    Fn wfn 5565   -->wf 5566   -onto->wfo 5568   ` cfv 5570   1stc1st 6771   2ndc2nd 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fo 5576  df-fv 5578  df-1st 6773  df-2nd 6774
This theorem is referenced by:  mbfmco2  28473
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