Theorem ndmaovcom 41285

Description: Any operation is commutative outside its domain, analogous to ndmovcom 6821. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypothesis

Ref	Expression
ndmaov.1	|- dom F = ( S X. S )

Assertion

Ref	Expression
ndmaovcom	|- ( -. ( A e. S /\ B e. S ) -> (( A F B)) = (( B F A)) )

Proof of Theorem ndmaovcom

Step	Hyp	Ref	Expression
1		opelxp 5146	. . . 4 |- ( <. A , B >. e. ( S X. S ) <-> ( A e. S /\ B e. S ) )
2		ndmaov.1	. . . . . 6 |- dom F = ( S X. S )
3	2	eqcomi 2631	. . . . 5 |- ( S X. S ) = dom F
4	3	eleq2i 2693	. . . 4 |- ( <. A , B >. e. ( S X. S ) <-> <. A , B >. e. dom F )
5	1, 4	bitr3i 266	. . 3 |- ( ( A e. S /\ B e. S ) <-> <. A , B >. e. dom F )
6		ndmaov 41263	. . 3 |- ( -. <. A , B >. e. dom F -> (( A F B)) = _V )
7	5, 6	sylnbi 320	. 2 |- ( -. ( A e. S /\ B e. S ) -> (( A F B)) = _V )
8		ancom 466	. . . 4 |- ( ( A e. S /\ B e. S ) <-> ( B e. S /\ A e. S ) )
9		opelxp 5146	. . . 4 |- ( <. B , A >. e. ( S X. S ) <-> ( B e. S /\ A e. S ) )
10	3	eleq2i 2693	. . . 4 |- ( <. B , A >. e. ( S X. S ) <-> <. B , A >. e. dom F )
11	8, 9, 10	3bitr2i 288	. . 3 |- ( ( A e. S /\ B e. S ) <-> <. B , A >. e. dom F )
12		ndmaov 41263	. . 3 |- ( -. <. B , A >. e. dom F -> (( B F A)) = _V )
13	11, 12	sylnbi 320	. 2 |- ( -. ( A e. S /\ B e. S ) -> (( B F A)) = _V )
14	7, 13	eqtr4d 2659	1 |- ( -. ( A e. S /\ B e. S ) -> (( A F B)) = (( B F A)) )

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-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)