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Theorem fndmdifcom 7046
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 2984 . . 3 ((𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐺𝑥) ≠ (𝐹𝑥))
21rabbii 3425 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = {𝑥𝐴 ∣ (𝐺𝑥) ≠ (𝐹𝑥)}
3 fndmdif 7045 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
4 fndmdif 7045 . . 3 ((𝐺 Fn 𝐴𝐹 Fn 𝐴) → dom (𝐺𝐹) = {𝑥𝐴 ∣ (𝐺𝑥) ≠ (𝐹𝑥)})
54ancoms 457 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐺𝐹) = {𝑥𝐴 ∣ (𝐺𝑥) ≠ (𝐹𝑥)})
62, 3, 53eqtr4a 2791 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wne 2930  {crab 3419  cdif 3937  dom cdm 5672   Fn wfn 6537  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by: (None)
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