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Theorem fnnfpeq0 6674
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 4158 . . 3 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) ≠ 𝑥)
2 fvresi 6669 . . . . . . 7 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
32eqeq2d 2810 . . . . . 6 (𝑥𝐴 → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
43adantl 474 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
5 nne 2976 . . . . 5 (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = 𝑥)
64, 5syl6rbbr 282 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
76ralbidva 3167 . . 3 (𝐹 Fn 𝐴 → (∀𝑥𝐴 ¬ (𝐹𝑥) ≠ 𝑥 ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
81, 7syl5bb 275 . 2 (𝐹 Fn 𝐴 → ({𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
9 fndifnfp 6672 . . 3 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
109eqeq1d 2802 . 2 (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅))
11 fnresi 6220 . . 3 ( I ↾ 𝐴) Fn 𝐴
12 eqfnfv 6538 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
1311, 12mpan2 683 . 2 (𝐹 Fn 𝐴 → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
148, 10, 133bitr4d 303 1 (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wne 2972  wral 3090  {crab 3094  cdif 3767  c0 4116   I cid 5220  dom cdm 5313  cres 5315   Fn wfn 6097  cfv 6102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fv 6110
This theorem is referenced by:  symggen  18201  m1detdiag  20728  mdetdiaglem  20729
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