Theorem fndmdifcom 5741

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 2484	. . . 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 2784	. 2 |- { x e. A | ( F ` x ) =/= ( G ` x ) } = { x e. A | ( G ` x ) =/= ( F ` x ) }
4		fndmdif 5740	. 2 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
5		fndmdif 5740	. . 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 2288	1 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = dom ( G \ F ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   =/= wne 2400   {crab 2512   \ cdif 3194   dom cdm 4719   Fn wfn 5313   ` cfv 5318

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by: (None)