Theorem fnnfpeq0 7152

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 4351	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ 𝑥)
2		nne 2929	. . . . 5 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) = 𝑥)
3		fvresi 7147	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
4	3	eqeq2d 2740	. . . . . 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 290	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ (𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
7	6	ralbidva 3154	. . 3 ⊢ (𝐹 Fn 𝐴 → (∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ 𝑥 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
8	1, 7	bitrid 283	. 2 ⊢ (𝐹 Fn 𝐴 → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
9		fndifnfp 7150	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
10	9	eqeq1d 2731	. 2 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅))
11		fnresi 6647	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
12		eqfnfv 7003	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
13	11, 12	mpan2 691	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
14	8, 10, 13	3bitr4d 311	1 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  {crab 3405   ∖ cdif 3911  ∅c0 4296   I cid 5532  dom cdm 5638   ↾ cres 5640   Fn wfn 6506  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519

This theorem is referenced by:  symggen  19400  m1detdiag  22484  mdetdiaglem  22485