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Theorem fvopab4ndm 5963
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 5195 . . . . 5  |-  dom  F  =  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
3 dmopabss 5205 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
42, 3eqsstri 3527 . . . 4  |-  dom  F  C_  A
54sseli 3493 . . 3  |-  ( B  e.  dom  F  ->  B  e.  A )
65con3i 135 . 2  |-  ( -.  B  e.  A  ->  -.  B  e.  dom  F )
7 ndmfv 5881 . 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 1374    e. wcel 1762   (/)c0 3778   {copab 4497   dom cdm 4992   ` cfv 5579
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-dm 5002  df-iota 5542  df-fv 5587
This theorem is referenced by: (None)
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