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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 (𝐹𝐺) = dom (𝐺𝐹))

Proof of Theorem fndmdifcom
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 necom 2283 . . . 4 ((𝐹x) ≠ (𝐺x) ↔ (𝐺x) ≠ (𝐹x))
21a1i 9 . . 3 (x A → ((𝐹x) ≠ (𝐺x) ↔ (𝐺x) ≠ (𝐹x)))
32rabbiia 2541 . 2 {x A ∣ (𝐹x) ≠ (𝐺x)} = {x A ∣ (𝐺x) ≠ (𝐹x)}
4 fndmdif 5215 . 2 ((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) = {x A ∣ (𝐹x) ≠ (𝐺x)})
5 fndmdif 5215 . . 3 ((𝐺 Fn A 𝐹 Fn A) → dom (𝐺𝐹) = {x A ∣ (𝐺x) ≠ (𝐹x)})
65ancoms 255 . 2 ((𝐹 Fn A 𝐺 Fn A) → dom (𝐺𝐹) = {x A ∣ (𝐺x) ≠ (𝐹x)})
73, 4, 63eqtr4a 2095 1 ((𝐹 Fn A 𝐺 Fn A) → dom (𝐹𝐺) = dom (𝐺𝐹))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ 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)
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