Theorem fndmdifcom 5784

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 2496	. . . 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 2799	. 2 |- { x e. A | ( F ` x ) =/= ( G ` x ) } = { x e. A | ( G ` x ) =/= ( F ` x ) }
4		fndmdif 5783	. 2 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = { x e. A | ( F ` x ) =/= ( G ` x ) } )
5		fndmdif 5783	. . 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 2291	1 |- ( ( F Fn A /\ G Fn A ) -> dom ( F \ G ) = dom ( G \ F ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   =/= wne 2412   {crab 2524   \ cdif 3208   dom cdm 4749   Fn wfn 5347   ` cfv 5352

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360

This theorem is referenced by: (None)