Theorem ndmaovcl 41283

Description: The "closure" of an operation outside its domain, when the operation's value is a set in contrast to ndmovcl 6819 where it is required that the domain contains the empty set ( (/) e. S). (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypotheses

Ref	Expression
ndmaov.1	|- dom F = ( S X. S )
ndmaovcl.2	|- ( ( A e. S /\ B e. S ) -> (( A F B)) e. S )
ndmaovcl.3	|- (( A F B)) e. _V

Assertion

Ref	Expression
ndmaovcl	|- (( A F B)) e. S

Proof of Theorem ndmaovcl

Step	Hyp	Ref	Expression
1		ndmaovcl.2	. 2 |- ( ( A e. S /\ B e. S ) -> (( A F B)) e. S )
2		opelxp 5146	. . 3 |- ( <. A , B >. e. ( S X. S ) <-> ( A e. S /\ B e. S ) )
3		ndmaov.1	. . . . . 6 |- dom F = ( S X. S )
4	3	eqcomi 2631	. . . . 5 |- ( S X. S ) = dom F
5	4	eleq2i 2693	. . . 4 |- ( <. A , B >. e. ( S X. S ) <-> <. A , B >. e. dom F )
6		ndmaovcl.3	. . . . 5 |- (( A F B)) e. _V
7		ndmaov 41263	. . . . 5 |- ( -. <. A , B >. e. dom F -> (( A F B)) = _V )
8		eleq1 2689	. . . . . . 7 |- ( (( A F B)) = _V -> ( (( A F B)) e. _V <-> _V e. _V ) )
9	8	biimpd 219	. . . . . 6 |- ( (( A F B)) = _V -> ( (( A F B)) e. _V -> _V e. _V ) )
10		vprc 4796	. . . . . . 7 |- -. _V e. _V
11	10	pm2.21i 116	. . . . . 6 |- ( _V e. _V -> (( A F B)) e. S )
12	9, 11	syl6com 37	. . . . 5 |- ( (( A F B)) e. _V -> ( (( A F B)) = _V -> (( A F B)) e. S ) )
13	6, 7, 12	mpsyl 68	. . . 4 |- ( -. <. A , B >. e. dom F -> (( A F B)) e. S )
14	5, 13	sylnbi 320	. . 3 |- ( -. <. A , B >. e. ( S X. S ) -> (( A F B)) e. S )
15	2, 14	sylnbir 321	. 2 |- ( -. ( A e. S /\ B e. S ) -> (( A F B)) e. S )
16	1, 15	pm2.61i 176	1 |- (( A F B)) e. S

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   X. cxp 5112   dom cdm 5114   ((caov 41195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-fv 5896  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)