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Theorem fvopab4ndm 5789
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 5036 . . . . 5  |-  dom  F  =  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
3 dmopabss 5046 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
42, 3eqsstri 3381 . . . 4  |-  dom  F  C_  A
54sseli 3347 . . 3  |-  ( B  e.  dom  F  ->  B  e.  A )
65con3i 135 . 2  |-  ( -.  B  e.  A  ->  -.  B  e.  dom  F )
7 ndmfv 5709 . 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 1369    e. wcel 1756   (/)c0 3632   {copab 4344   dom cdm 4835   ` cfv 5413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-dm 4845  df-iota 5376  df-fv 5421
This theorem is referenced by: (None)
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