Theorem fnnfpeq0 6917

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 4292	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ 𝑥)
2		nne 2991	. . . . 5 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) = 𝑥)
3		fvresi 6912	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
4	3	eqeq2d 2809	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) = 𝑥))
5	4	adantl 485	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) = 𝑥))
6	2, 5	bitr4id 293	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (¬ (𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
7	6	ralbidva 3161	. . 3 ⊢ (𝐹 Fn 𝐴 → (∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ 𝑥 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
8	1, 7	syl5bb 286	. 2 ⊢ (𝐹 Fn 𝐴 → ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
9		fndifnfp 6915	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
10	9	eqeq1d 2800	. 2 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅))
11		fnresi 6448	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
12		eqfnfv 6779	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
13	11, 12	mpan2 690	. 2 ⊢ (𝐹 Fn 𝐴 → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
14	8, 10, 13	3bitr4d 314	1 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  {crab 3110   ∖ cdif 3878  ∅c0 4243   I cid 5424  dom cdm 5519   ↾ cres 5521   Fn wfn 6319  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332

This theorem is referenced by:  symggen  18590  m1detdiag  21202  mdetdiaglem  21203