Theorem fvopab4ndm 3798

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 3326	. . . . 5 |- dom F = dom {<.x, y>. | (x e. A /\ ph)}
3		dmopabss 3335	. . . . 5 |- dom {<.x, y>. | (x e. A /\ ph)} (_ A
4	2, 3	eqsstr 2100	. . . 4 |- dom F (_ A
5	4	sseli 2074	. . 3 |- (B e. dom F -> B e. A)
6	5	con3i 98	. 2 |- (-. B e. A -> -. B e. dom F)
7		ndmfv 3759	. 2 |- (-. B e. dom F -> (F` B) = (/))
8	6, 7	syl 10	1 |- (-. B e. A -> (F` B) = (/))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 960   e. wcel 962  (/)c0 2289  {copab 2679  dom cdm 3184  ` cfv 3196

This theorem is referenced by:  curry1val 4114

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-sep 2716  ax-pow 2756  ax-pr 2793

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-rex 1657  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-xp 3198  df-cnv 3200  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fv 3212

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