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Theorem fnnfpeq0 6588
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 4103 . . 3 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) ≠ 𝑥)
2 fvresi 6583 . . . . . . 7 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
32eqeq2d 2781 . . . . . 6 (𝑥𝐴 → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
43adantl 467 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
5 nne 2947 . . . . 5 (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = 𝑥)
64, 5syl6rbbr 279 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
76ralbidva 3134 . . 3 (𝐹 Fn 𝐴 → (∀𝑥𝐴 ¬ (𝐹𝑥) ≠ 𝑥 ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
81, 7syl5bb 272 . 2 (𝐹 Fn 𝐴 → ({𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
9 fndifnfp 6586 . . 3 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
109eqeq1d 2773 . 2 (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅))
11 fnresi 6148 . . 3 ( I ↾ 𝐴) Fn 𝐴
12 eqfnfv 6454 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
1311, 12mpan2 671 . 2 (𝐹 Fn 𝐴 → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
148, 10, 133bitr4d 300 1 (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  {crab 3065  cdif 3720  c0 4063   I cid 5156  dom cdm 5249  cres 5251   Fn wfn 6026  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039
This theorem is referenced by:  symggen  18097  m1detdiag  20621  mdetdiaglem  20622
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