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Theorem fvopab4ndm 5959
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 5190 . . . . 5  |-  dom  F  =  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }
3 dmopabss 5200 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  C_  A
42, 3eqsstri 3516 . . . 4  |-  dom  F  C_  A
54sseli 3482 . . 3  |-  ( B  e.  dom  F  ->  B  e.  A )
65con3i 135 . 2  |-  ( -.  B  e.  A  ->  -.  B  e.  dom  F )
7 ndmfv 5876 . 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 1381    e. wcel 1802   (/)c0 3767   {copab 4490   dom cdm 4985   ` cfv 5574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-dm 4995  df-iota 5537  df-fv 5582
This theorem is referenced by: (None)
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