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Theorem fnnfpeq0 7050
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 4318 . . 3 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) ≠ 𝑥)
2 nne 2947 . . . . 5 (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = 𝑥)
3 fvresi 7045 . . . . . . 7 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
43eqeq2d 2749 . . . . . 6 (𝑥𝐴 → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
54adantl 482 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
62, 5bitr4id 290 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
76ralbidva 3111 . . 3 (𝐹 Fn 𝐴 → (∀𝑥𝐴 ¬ (𝐹𝑥) ≠ 𝑥 ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
81, 7bitrid 282 . 2 (𝐹 Fn 𝐴 → ({𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
9 fndifnfp 7048 . . 3 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
109eqeq1d 2740 . 2 (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅))
11 fnresi 6561 . . 3 ( I ↾ 𝐴) Fn 𝐴
12 eqfnfv 6909 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
1311, 12mpan2 688 . 2 (𝐹 Fn 𝐴 → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
148, 10, 133bitr4d 311 1 (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  {crab 3068  cdif 3884  c0 4256   I cid 5488  dom cdm 5589  cres 5591   Fn wfn 6428  cfv 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441
This theorem is referenced by:  symggen  19078  m1detdiag  21746  mdetdiaglem  21747
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