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Theorem fnnfpeq0 7125
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 4345 . . 3 ({𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥𝐴 ¬ (𝐹𝑥) ≠ 𝑥)
2 nne 2944 . . . . 5 (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = 𝑥)
3 fvresi 7120 . . . . . . 7 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
43eqeq2d 2744 . . . . . 6 (𝑥𝐴 → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
54adantl 483 . . . . 5 ((𝐹 Fn 𝐴𝑥𝐴) → ((𝐹𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) = 𝑥))
62, 5bitr4id 290 . . . 4 ((𝐹 Fn 𝐴𝑥𝐴) → (¬ (𝐹𝑥) ≠ 𝑥 ↔ (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
76ralbidva 3169 . . 3 (𝐹 Fn 𝐴 → (∀𝑥𝐴 ¬ (𝐹𝑥) ≠ 𝑥 ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
81, 7bitrid 283 . 2 (𝐹 Fn 𝐴 → ({𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅ ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
9 fndifnfp 7123 . . 3 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
109eqeq1d 2735 . 2 (𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥} = ∅))
11 fnresi 6631 . . 3 ( I ↾ 𝐴) Fn 𝐴
12 eqfnfv 6983 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → (𝐹 = ( I ↾ 𝐴) ↔ ∀𝑥𝐴 (𝐹𝑥) = (( I ↾ 𝐴)‘𝑥)))
1311, 12mpan2 690 . 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 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  {crab 3406  cdif 3908  c0 4283   I cid 5531  dom cdm 5634  cres 5636   Fn wfn 6492  cfv 6497
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505
This theorem is referenced by:  symggen  19257  m1detdiag  21962  mdetdiaglem  21963
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