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Theorem fvopab4ndm 5896
Description: Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
Hypothesis
Ref Expression
fvopab4ndm.1  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }
Assertion
Ref Expression
fvopab4ndm  |-  ( -.  B  e.  A  -> 
( F `  B
)  =  (/) )
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)    B( x, y)    F( x, y)

Proof of Theorem fvopab4ndm
StepHypRef Expression
1 fvopab4ndm.1 . . . . . 6  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }
21dmeqi 5142 . . . . 5  |-  dom  F  =  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
3 dmopabss 5152 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
42, 3eqsstri 3487 . . . 4  |-  dom  F  C_  A
54sseli 3453 . . 3  |-  ( B  e.  dom  F  ->  B  e.  A )
65con3i 135 . 2  |-  ( -.  B  e.  A  ->  -.  B  e.  dom  F )
7 ndmfv 5816 . 2  |-  ( -.  B  e.  dom  F  ->  ( F `  B
)  =  (/) )
86, 7syl 16 1  |-  ( -.  B  e.  A  -> 
( F `  B
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   (/)c0 3738   {copab 4450   dom cdm 4941   ` cfv 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-dm 4951  df-iota 5482  df-fv 5527
This theorem is referenced by: (None)
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