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Theorem fndmdifcom 6230
 Description: The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmdifcom ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom (𝐺𝐹))

Proof of Theorem fndmdifcom
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 necom 2835 . . . 4 ((𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐺𝑥) ≠ (𝐹𝑥))
21a1i 11 . . 3 (𝑥𝐴 → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐺𝑥) ≠ (𝐹𝑥)))
32rabbiia 3161 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = {𝑥𝐴 ∣ (𝐺𝑥) ≠ (𝐹𝑥)}
4 fndmdif 6229 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
5 fndmdif 6229 . . 3 ((𝐺 Fn 𝐴𝐹 Fn 𝐴) → dom (𝐺𝐹) = {𝑥𝐴 ∣ (𝐺𝑥) ≠ (𝐹𝑥)})
65ancoms 468 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐺𝐹) = {𝑥𝐴 ∣ (𝐺𝑥) ≠ (𝐹𝑥)})
73, 4, 63eqtr4a 2670 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom (𝐺𝐹))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  {crab 2900   ∖ cdif 3537  dom cdm 5038   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by: (None)
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