Theorem fndmdifcom 5665

Description: The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)

Assertion

Ref	Expression
fndmdifcom	|- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = dom ( G \ F ) )

Proof of Theorem fndmdifcom

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		necom 2448	. . . 4 |- ( ( F ` x ) =/= ( G ` x ) <-> ( G ` x ) =/= ( F ` x ) )
2	1	a1i 9	. . 3 |- ( x e. A -> ( ( F ` x ) =/= ( G ` x ) <-> ( G ` x ) =/= ( F ` x ) ) )
3	2	rabbiia 2745	. 2 |- { x e. A | ( F ` x ) =/= ( G ` x ) } = { x e. A | ( G ` x ) =/= ( F ` x ) }
4		fndmdif 5664	. 2 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
5		fndmdif 5664	. . 3 |- ( ( G Fn A /\ F Fn A ) -> dom ( G \ F ) = { x e. A | ( G ` x ) =/= ( F ` x ) } )
6	5	ancoms 268	. 2 |- ( ( F Fn A /\ G Fn A ) -> dom ( G \ F ) = { x e. A | ( G ` x ) =/= ( F ` x ) } )
7	3, 4, 6	3eqtr4a 2252	1 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = dom ( G \ F ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   =/= wne 2364   {crab 2476   \ cdif 3151   dom cdm 4660   Fn wfn 5250   ` cfv 5255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263

This theorem is referenced by: (None)