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Theorem fndmdifeq0 7063
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 7061 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
21eqeq1d 2736 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅))
3 rabeq0 4393 . . . 4 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) ≠ (𝐺𝑥))
4 nne 2941 . . . . 5 (¬ (𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐹𝑥) = (𝐺𝑥))
54ralbii 3090 . . . 4 (∀𝑥𝐴 ¬ (𝐹𝑥) ≠ (𝐺𝑥) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
63, 5bitri 275 . . 3 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
7 eqfnfv 7050 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
86, 7bitr4id 290 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ 𝐹 = 𝐺))
92, 8bitrd 279 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ 𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wne 2937  wral 3058  {crab 3432  cdif 3959  c0 4338  dom cdm 5688   Fn wfn 6557  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570
This theorem is referenced by:  wemapso  9588  wemapso2lem  9589
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