Theorem fvopab4ndm 5803

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

Step	Hyp	Ref	Expression
1		fvopab4ndm.1	. . . . 5 |- F = { <. x , y >. | ( x e. A /\ ph ) }
2	1	dmeqi 4982	. . . 4 |- dom F = dom { <. x , y >. | ( x e. A /\ ph ) }
3		dmopabss 4993	. . . 4 |- dom { <. x , y >. | ( x e. A /\ ph ) } C_ A
4	2, 3	eqsstri 3280	. . 3 |- dom F C_ A
5	4	sseli 3244	. 2 |- ( B e. dom F -> B e. A )
6	1	relopabiv 4903	. . 3 |- Rel F
7		relndmfv 5728	. . 3 |- ( ( Rel F /\ -. B e. dom F ) -> ( F ` B ) = (/) )
8	6, 7	mpan 428	. 2 |- ( -. B e. dom F -> ( F ` B ) = (/) )
9	5, 8	nsyl5 659	1 |- ( -. B e. A -> ( F ` B ) = (/) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   (/)c0 3520   {copab 4191   dom cdm 4774   Rel wrel 4779   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-dm 4784  df-iota 5337  df-fv 5385

This theorem is used by:  fvmptndm  5804