Theorem ndmoprcl 4977

Description: The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is is dependent on our particular definitions of operation value, function value, and ordered pair.

Hypotheses

Ref	Expression
ndmoprcl.1	|- dom F = (S X. S)
ndmoprcl.2	|- ((A e. S /\ x e. S) -> (AFx) e. S)
ndmoprcl.3	|- (/) e. S

Assertion

Ref	Expression
ndmoprcl	|- (AFB) e. S

Distinct variable groups:   x,A   x,B   x,F   x,S

Proof of Theorem ndmoprcl

Step	Hyp	Ref	Expression
1		oprprc2 4909	. . . . 5 |- (-. B e. _V -> (AFB) = (AFA))
2	1	eleq1d 1963	. . . 4 |- (-. B e. _V -> ((AFB) e. S <-> (AFA) e. S))
3		ndmoprcl.1	. . . . . . . 8 |- dom F = (S X. S)
4		ndmoprg 4976	. . . . . . . 8 |- ((dom F = (S X. S) /\ A e. _V /\ -. (A e. S /\ A e. S)) -> (AFA) = (/))
5	3, 4	mp3an1 1178	. . . . . . 7 |- ((A e. _V /\ -. (A e. S /\ A e. S)) -> (AFA) = (/))
6		ndmoprcl.3	. . . . . . 7 |- (/) e. S
7	5, 6	syl6eqel 1979	. . . . . 6 |- ((A e. _V /\ -. (A e. S /\ A e. S)) -> (AFA) e. S)
8	7	ex 402	. . . . 5 |- (A e. _V -> (-. (A e. S /\ A e. S) -> (AFA) e. S))
9		opreq2 4890	. . . . . . . . 9 |- (x = A -> (AFx) = (AFA))
10	9	eleq1d 1963	. . . . . . . 8 |- (x = A -> ((AFx) e. S <-> (AFA) e. S))
11	10	imbi2d 674	. . . . . . 7 |- (x = A -> ((A e. S -> (AFx) e. S) <-> (A e. S -> (AFA) e. S)))
12		ndmoprcl.2	. . . . . . . 8 |- ((A e. S /\ x e. S) -> (AFx) e. S)
13	12	expcom 403	. . . . . . 7 |- (x e. S -> (A e. S -> (AFx) e. S))
14	11, 13	vtoclga 2352	. . . . . 6 |- (A e. S -> (A e. S -> (AFA) e. S))
15	14	imp 377	. . . . 5 |- ((A e. S /\ A e. S) -> (AFA) e. S)
16	8, 15	pm2.61d2 143	. . . 4 |- (A e. _V -> (AFA) e. S)
17	2, 16	syl5cbir 228	. . 3 |- (A e. _V -> (-. B e. _V -> (AFB) e. S))
18		ndmoprg 4976	. . . . . . 7 |- ((dom F = (S X. S) /\ B e. _V /\ -. (A e. S /\ B e. S)) -> (AFB) = (/))
19	3, 18	mp3an1 1178	. . . . . 6 |- ((B e. _V /\ -. (A e. S /\ B e. S)) -> (AFB) = (/))
20	19, 6	syl6eqel 1979	. . . . 5 |- ((B e. _V /\ -. (A e. S /\ B e. S)) -> (AFB) e. S)
21	20	ex 402	. . . 4 |- (B e. _V -> (-. (A e. S /\ B e. S) -> (AFB) e. S))
22		opreq2 4890	. . . . . . . 8 |- (x = B -> (AFx) = (AFB))
23	22	eleq1d 1963	. . . . . . 7 |- (x = B -> ((AFx) e. S <-> (AFB) e. S))
24	23	imbi2d 674	. . . . . 6 |- (x = B -> ((A e. S -> (AFx) e. S) <-> (A e. S -> (AFB) e. S)))
25	24, 13	vtoclga 2352	. . . . 5 |- (B e. S -> (A e. S -> (AFB) e. S))
26	25	impcom 378	. . . 4 |- ((A e. S /\ B e. S) -> (AFB) e. S)
27	21, 26	pm2.61d2 143	. . 3 |- (B e. _V -> (AFB) e. S)
28	17, 27	pm2.61d2 143	. 2 |- (A e. _V -> (AFB) e. S)
29		relxp 4088	. . . . 5 |- Rel (S X. S)
30	3	releqi 4072	. . . . 5 |- (Rel dom F <-> Rel (S X. S))
31	29, 30	mpbir 207	. . . 4 |- Rel dom F
32	31	oprprc1 4908	. . 3 |- (-. A e. _V -> (AFB) = (/))
33	32, 6	syl6eqel 1979	. 2 |- (-. A e. _V -> (AFB) e. S)
34	28, 33	pm2.61i 140	1 |- (AFB) e. S

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  (/)c0 2875   X. cxp 3984  dom cdm 3986  Rel wrel 3991  (class class class)co 4884

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-3an 860  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-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886

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