Theorem fvopab4ndm 4747

Description: Value of a function given by an ordered-pair class abstraction, outside of its domain.

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

Proof of Theorem fvopab4ndm

Step	Hyp	Ref	Expression
1		fvopab4ndm.1	. . . . . 6 |- F = {<.x, y>. | (x e. A /\ ph)}
2	1	dmeqi 4158	. . . . 5 |- dom F = dom {<.x, y>. | (x e. A /\ ph)}
3		dmopabss 4168	. . . . 5 |- dom {<.x, y>. | (x e. A /\ ph)} C_ A
4	2, 3	eqsstri 2647	. . . 4 |- dom F C_ A
5	4	sseli 2617	. . 3 |- (B e. dom F -> B e. A)
6	5	con3i 114	. 2 |- (-. B e. A -> -. B e. dom F)
7		ndmfv 4702	. 2 |- (-. B e. dom F -> (F` B) = (/))
8	6, 7	syl 12	1 |- (-. B e. A -> (F` B) = (/))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  (/)c0 2875  {copab 3395  dom cdm 3986  ` cfv 3998

This theorem is referenced by:  curry1val 5077  curry2val 5080

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014

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