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Theorem fndmdifeq0 6915
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 6913 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
21eqeq1d 2741 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅))
3 rabeq0 4323 . . . 4 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) ≠ (𝐺𝑥))
4 nne 2948 . . . . 5 (¬ (𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐹𝑥) = (𝐺𝑥))
54ralbii 3092 . . . 4 (∀𝑥𝐴 ¬ (𝐹𝑥) ≠ (𝐺𝑥) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
63, 5bitri 274 . . 3 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
7 eqfnfv 6903 . . 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 395   = wceq 1541  wne 2944  wral 3065  {crab 3069  cdif 3888  c0 4261  dom cdm 5588   Fn wfn 6425  cfv 6430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-fv 6438
This theorem is referenced by:  wemapso  9271  wemapso2lem  9272
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