Theorem fndmdifcom 6971

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 2979	. . 3 ⊢ ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ (𝐹‘𝑥))
2	1	rabbii 3398	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)}
3		fndmdif 6970	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})
4		fndmdif 6970	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐹 Fn 𝐴) → dom (𝐺 ∖ 𝐹) = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)})
5	4	ancoms 458	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐺 ∖ 𝐹) = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)})
6	2, 3, 5	3eqtr4a 2791	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = dom (𝐺 ∖ 𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ≠ wne 2926  {crab 3393   ∖ cdif 3897  dom cdm 5614   Fn wfn 6472  ‘cfv 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fn 6480  df-fv 6485

This theorem is referenced by: (None)