Theorem fnnfpeq0 7127

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 4329	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥 ∈ 𝐴 ¬ (𝐹‘𝑥) ≠ 𝑥)
2		nne 2937	. . . . 5 ⊢ (¬ (𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑥) = 𝑥)
3		fvresi 7122	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
4	3	eqeq2d 2748	. . . . . 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 3159	. . 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 7125	. . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
10	9	eqeq1d 2739	. 2 ⊢ (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥} = ∅))
11		fnresi 6622	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
12		eqfnfv 6978	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)))
13	11, 12	mpan2 692	. 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 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3390   ∖ cdif 3887  ∅c0 4274   I cid 5519  dom cdm 5625   ↾ cres 5627   Fn wfn 6488  ‘cfv 6493

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501

This theorem is referenced by:  symggen  19439  m1detdiag  22575  mdetdiaglem  22576