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Theorem fvopab4ndm 5620
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 4880 . . . . 5  |-  dom  F  =  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
3 dmopabss 4890 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
42, 3eqsstri 3208 . . . 4  |-  dom  F  C_  A
54sseli 3176 . . 3  |-  ( B  e.  dom  F  ->  B  e.  A )
65con3i 127 . 2  |-  ( -.  B  e.  A  ->  -.  B  e.  dom  F )
7 ndmfv 5552 . 2  |-  ( -.  B  e.  dom  F  ->  ( F `  B
)  =  (/) )
86, 7syl 15 1  |-  ( -.  B  e.  A  -> 
( F `  B
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   (/)c0 3455   {copab 4076   dom cdm 4689   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-dm 4699  df-iota 5219  df-fv 5263
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