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Theorem fndmdifcom 5625
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 2431 . . . 4 ((𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐺𝑥) ≠ (𝐹𝑥))
21a1i 9 . . 3 (𝑥𝐴 → ((𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐺𝑥) ≠ (𝐹𝑥)))
32rabbiia 2724 . 2 {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = {𝑥𝐴 ∣ (𝐺𝑥) ≠ (𝐹𝑥)}
4 fndmdif 5624 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
5 fndmdif 5624 . . 3 ((𝐺 Fn 𝐴𝐹 Fn 𝐴) → dom (𝐺𝐹) = {𝑥𝐴 ∣ (𝐺𝑥) ≠ (𝐹𝑥)})
65ancoms 268 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐺𝐹) = {𝑥𝐴 ∣ (𝐺𝑥) ≠ (𝐹𝑥)})
73, 4, 63eqtr4a 2236 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom (𝐺𝐹))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wne 2347  {crab 2459  cdif 3128  dom cdm 4628   Fn wfn 5213  cfv 5218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226
This theorem is referenced by: (None)
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