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Theorem fndmdifeq0 6549
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 6547 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
21eqeq1d 2801 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅))
3 eqfnfv 6537 . . 3 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
4 rabeq0 4157 . . . 4 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) ≠ (𝐺𝑥))
5 nne 2975 . . . . 5 (¬ (𝐹𝑥) ≠ (𝐺𝑥) ↔ (𝐹𝑥) = (𝐺𝑥))
65ralbii 3161 . . . 4 (∀𝑥𝐴 ¬ (𝐹𝑥) ≠ (𝐺𝑥) ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
74, 6bitri 267 . . 3 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))
83, 7syl6rbbr 282 . 2 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ({𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)} = ∅ ↔ 𝐹 = 𝐺))
92, 8bitrd 271 1 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ 𝐹 = 𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wne 2971  wral 3089  {crab 3093  cdif 3766  c0 4115  dom cdm 5312   Fn wfn 6096  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109
This theorem is referenced by:  wemapso  8698  wemapso2lem  8699
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