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Theorem fndmdifeq0 7057
Description: The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmdifeq0 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ 𝐹 = 𝐺))

Proof of Theorem fndmdifeq0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fndmdif 7055 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
21eqeq1d 2728 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅))
3 rabeq0 4389 . . . 4 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) ≠ (𝐺𝑥))
4 nne 2934 . . . . 5 (¬ (𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐹𝑥) = (𝐺𝑥))
54ralbii 3083 . . . 4 (∀𝑥𝐴 ¬ (𝐹𝑥) ≠ (𝐺𝑥) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
63, 5bitri 274 . . 3 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
7 eqfnfv 7044 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
86, 7bitr4id 289 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ 𝐹 = 𝐺))
92, 8bitrd 278 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ 𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394   = wceq 1534  wne 2930  wral 3051  {crab 3419  cdif 3944  c0 4325  dom cdm 5682   Fn wfn 6549  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-fv 6562
This theorem is referenced by:  wemapso  9594  wemapso2lem  9595
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