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Theorem fndmdifeq0 6279
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 6277 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
21eqeq1d 2623 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅))
3 eqfnfv 6267 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
4 rabeq0 3931 . . . 4 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) ≠ (𝐺𝑥))
5 nne 2794 . . . . 5 (¬ (𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐹𝑥) = (𝐺𝑥))
65ralbii 2974 . . . 4 (∀𝑥𝐴 ¬ (𝐹𝑥) ≠ (𝐺𝑥) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
74, 6bitri 264 . . 3 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
83, 7syl6rbbr 279 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ 𝐹 = 𝐺))
92, 8bitrd 268 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ 𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wne 2790  wral 2907  {crab 2911  cdif 3552  c0 3891  dom cdm 5074   Fn wfn 5842  cfv 5847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855
This theorem is referenced by:  wemapso  8400  wemapso2lem  8401
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