Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnnfpeq0 Structured version   Visualization version   GIF version

Theorem fnnfpeq0 6349
 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.
StepHypRef Expression
1 rabeq0 3911 . . 3 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) ≠ 𝑥)
2 fvresi 6344 . . . . . . 7 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
32eqeq2d 2620 . . . . . 6 (𝑥𝐴 → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
43adantl 481 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
5 nne 2786 . . . . 5 (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = 𝑥)
64, 5syl6rbbr 278 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
76ralbidva 2968 . . 3 (𝐹 Fn 𝐴 → (∀𝑥𝐴 ¬ (𝐹𝑥) ≠ 𝑥 ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
81, 7syl5bb 271 . 2 (𝐹 Fn 𝐴 → ({𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
9 fndifnfp 6347 . . 3 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
109eqeq1d 2612 . 2 (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅))
11 fnresi 5922 . . 3 ( I ↾ 𝐴) Fn 𝐴
12 eqfnfv 6219 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
1311, 12mpan2 703 . 2 (𝐹 Fn 𝐴 → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
148, 10, 133bitr4d 299 1 (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900   ∖ cdif 3537  ∅c0 3874   I cid 4948  dom cdm 5038   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812 This theorem is referenced by:  symggen  17713  m1detdiag  20222  mdetdiaglem  20223
 Copyright terms: Public domain W3C validator