Theorem fndmdifcom 5216

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 2283	. . . 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 2541	. 2 |- {x e. A | ( F ` x) =/= ( G ` x) } = {x e. A | ( G ` x) =/= ( F ` x) }
4		fndmdif 5215	. 2 |- (( F Fn A /\ G Fn A) -> dom ( F \ G) = {x e. A | ( F ` x) =/= ( G ` x) })
5		fndmdif 5215	. . 3 |- (( G Fn A /\ F Fn A) -> dom ( G \ F) = {x e. A | ( G ` x) =/= ( F ` x) })
6	5	ancoms 255	. 2 |- (( F Fn A /\ G Fn A) -> dom ( G \ F) = {x e. A | ( G ` x) =/= ( F ` x) })
7	3, 4, 6	3eqtr4a 2095	1 |- (( F Fn A /\ G Fn A) -> dom ( F \ G) = dom ( G \ F))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   e. wcel 1390   =/= wne 2201   {crab 2304   \ cdif 2908   dom cdm 4288   Fn wfn 4840   ` cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by: (None)