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Theorem aovmpt4g 38093
Description: Value of a function given by the "maps to" notation, analogous to ovmpt4g 6433. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
aovmpt4g.3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
aovmpt4g  |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  -> (( x F y))  =  C )
Distinct variable groups:    x, y, A    x, B, y    x, C, y    x, V, y
Allowed substitution hints:    F( x, y)

Proof of Theorem aovmpt4g
StepHypRef Expression
1 aovmpt4g.3 . . . . . . 7  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
21dmmpt2g 6880 . . . . . 6  |-  ( C  e.  V  ->  dom  F  =  ( A  X.  B ) )
3 opelxpi 4886 . . . . . . 7  |-  ( ( x  e.  A  /\  y  e.  B )  -> 
<. x ,  y >.  e.  ( A  X.  B
) )
4 eleq2 2502 . . . . . . 7  |-  ( dom 
F  =  ( A  X.  B )  -> 
( <. x ,  y
>.  e.  dom  F  <->  <. x ,  y >.  e.  ( A  X.  B ) ) )
53, 4syl5ibr 224 . . . . . 6  |-  ( dom 
F  =  ( A  X.  B )  -> 
( ( x  e.  A  /\  y  e.  B )  ->  <. x ,  y >.  e.  dom  F ) )
62, 5syl 17 . . . . 5  |-  ( C  e.  V  ->  (
( x  e.  A  /\  y  e.  B
)  ->  <. x ,  y >.  e.  dom  F ) )
76impcom 431 . . . 4  |-  ( ( ( x  e.  A  /\  y  e.  B
)  /\  C  e.  V )  ->  <. x ,  y >.  e.  dom  F )
873impa 1200 . . 3  |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  -> 
<. x ,  y >.  e.  dom  F )
91mpt2fun 6412 . . . 4  |-  Fun  F
10 funres 5640 . . . 4  |-  ( Fun 
F  ->  Fun  ( F  |`  { <. x ,  y
>. } ) )
119, 10ax-mp 5 . . 3  |-  Fun  ( F  |`  { <. x ,  y >. } )
12 df-dfat 38008 . . . 4  |-  ( F defAt  <. x ,  y >.  <->  (
<. x ,  y >.  e.  dom  F  /\  Fun  ( F  |`  { <. x ,  y >. } ) ) )
13 aovfundmoveq 38073 . . . 4  |-  ( F defAt  <. x ,  y >.  -> (( x F y))  =  ( x F y ) )
1412, 13sylbir 216 . . 3  |-  ( (
<. x ,  y >.  e.  dom  F  /\  Fun  ( F  |`  { <. x ,  y >. } ) )  -> (( x F
y))  =  ( x F y ) )
158, 11, 14sylancl 666 . 2  |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  -> (( x F y))  =  ( x F y ) )
161ovmpt4g 6433 . 2  |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  ->  ( x F y )  =  C )
1715, 16eqtrd 2470 1  |-  ( ( x  e.  A  /\  y  e.  B  /\  C  e.  V )  -> (( x F y))  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {csn 4002   <.cop 4008    X. cxp 4852   dom cdm 4854    |` cres 4856   Fun wfun 5595  (class class class)co 6305    |-> cmpt2 6307   defAt wdfat 38005   ((caov 38007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-dfat 38008  df-afv 38009  df-aov 38010
This theorem is referenced by: (None)
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