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 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∖ 𝐺) = ∅ ↔ 𝐹 = 𝐺))

Proof of Theorem fndmdifeq0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fndmdif 6913	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})
2	1	eqeq1d 2741	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∖ 𝐺) = ∅ ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = ∅))
3		rabeq0 4323	. . . 4 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ (𝐺‘𝑥))
4		nne 2948	. . . . 5 ⊢ (¬ (𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ (𝐹‘𝑥) = (𝐺‘𝑥))
5	4	ralbii 3092	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ (𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))
6	3, 5	bitri 274	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = ∅ ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))
7		eqfnfv 6903	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
8	6, 7	bitr4id 289	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)} = ∅ ↔ 𝐹 = 𝐺))
9	2, 8	bitrd 278	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∖ 𝐺) = ∅ ↔ 𝐹 = 𝐺))

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