Theorem fndmdifcom 6322

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 2847	. . . 4 |- ( ( F ` x ) =/= ( G ` x ) <-> ( G ` x ) =/= ( F ` x ) )
2	1	a1i 11	. . 3 |- ( x e. A -> ( ( F ` x ) =/= ( G ` x ) <-> ( G ` x ) =/= ( F ` x ) ) )
3	2	rabbiia 3185	. 2 |- { x e. A | ( F ` x ) =/= ( G ` x ) } = { x e. A | ( G ` x ) =/= ( F ` x ) }
4		fndmdif 6321	. 2 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
5		fndmdif 6321	. . 3 |- ( ( G Fn A /\ F Fn A ) -> dom ( G \ F ) = { x e. A | ( G ` x ) =/= ( F ` x ) } )
6	5	ancoms 469	. 2 |- ( ( F Fn A /\ G Fn A ) -> dom ( G \ F ) = { x e. A | ( G ` x ) =/= ( F ` x ) } )
7	3, 4, 6	3eqtr4a 2682	1 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = dom ( G \ F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   \ cdif 3571   dom cdm 5114   Fn wfn 5883   ` cfv 5888

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by: (None)