Theorem fnnfpeq0 7179

Description: A function is the identity iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.)

Assertion

Ref	Expression
fnnfpeq0	⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴)))

Proof of Theorem fnnfpeq0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabeq0 4385	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ 𝑥)
2		nne 2943	. . . . 5 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) = 𝑥)
3		fvresi 7174	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
4	3	eqeq2d 2742	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) = 𝑥))
5	4	adantl 481	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) = 𝑥))
6	2, 5	bitr4id 289	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ (𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
7	6	ralbidva 3174	. . 3 ⊢ (𝐹 Fn 𝐴 → (∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ 𝑥 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
8	1, 7	bitrid 282	. 2 ⊢ (𝐹 Fn 𝐴 → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
9		fndifnfp 7177	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
10	9	eqeq1d 2733	. 2 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅))
11		fnresi 6680	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
12		eqfnfv 7033	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
13	11, 12	mpan2 688	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
14	8, 10, 13	3bitr4d 310	1 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  {crab 3431   ∖ cdif 3946  ∅c0 4323   I cid 5574  dom cdm 5677   ↾ cres 5679   Fn wfn 6539  ‘cfv 6544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552

This theorem is referenced by:  symggen  19380  m1detdiag  22320  mdetdiaglem  22321