Theorem fndmdifcom 5294

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 2329	. . . 4 ⊢ ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ (𝐹‘𝑥))
2	1	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐺‘𝑥) ≠ (𝐹‘𝑥)))
3	2	rabbiia 2591	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)}
4		fndmdif 5293	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})
5		fndmdif 5293	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝐹 Fn 𝐴) → dom (𝐺 ∖ 𝐹) = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)})
6	5	ancoms 264	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐺 ∖ 𝐹) = {𝑥 ∈ 𝐴 ∣ (𝐺‘𝑥) ≠ (𝐹‘𝑥)})
7	3, 4, 6	3eqtr4a 2139	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = dom (𝐺 ∖ 𝐹))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433   ≠ wne 2245  {crab 2352   ∖ cdif 2970  dom cdm 4363   Fn wfn 4917  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by: (None)