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Theorem fndifnfp 7196
Description: Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
Assertion
Ref Expression
fndifnfp (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem fndifnfp
StepHypRef Expression
1 dffn2 6739 . . . . . . . 8 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
2 fssxp 6764 . . . . . . . 8 (𝐹:𝐴⟶V → 𝐹 ⊆ (𝐴 × V))
31, 2sylbi 217 . . . . . . 7 (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × V))
4 ssdif0 4372 . . . . . . 7 (𝐹 ⊆ (𝐴 × V) ↔ (𝐹 ∖ (𝐴 × V)) = ∅)
53, 4sylib 218 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹 ∖ (𝐴 × V)) = ∅)
65uneq2d 4178 . . . . 5 (𝐹 Fn 𝐴 → ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ ∅))
7 un0 4400 . . . . 5 ((𝐹 ∖ I ) ∪ ∅) = (𝐹 ∖ I )
86, 7eqtr2di 2792 . . . 4 (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))))
9 df-res 5701 . . . . . 6 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
109difeq2i 4133 . . . . 5 (𝐹 ∖ ( I ↾ 𝐴)) = (𝐹 ∖ ( I ∩ (𝐴 × V)))
11 difindi 4298 . . . . 5 (𝐹 ∖ ( I ∩ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
1210, 11eqtri 2763 . . . 4 (𝐹 ∖ ( I ↾ 𝐴)) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
138, 12eqtr4di 2793 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = (𝐹 ∖ ( I ↾ 𝐴)))
1413dmeqd 5919 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ ( I ↾ 𝐴)))
15 fnresi 6698 . . 3 ( I ↾ 𝐴) Fn 𝐴
16 fndmdif 7062 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
1715, 16mpan2 691 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
18 fvresi 7193 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
1918neeq2d 2999 . . . 4 (𝑥𝐴 → ((𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) ≠ 𝑥))
2019rabbiia 3437 . . 3 {𝑥𝐴 ∣ (𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥)} = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥}
2120a1i 11 . 2 (𝐹 Fn 𝐴 → {𝑥𝐴 ∣ (𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥)} = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
2214, 17, 213eqtrd 2779 1 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wne 2938  {crab 3433  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339   I cid 5582   × cxp 5687  dom cdm 5689  cres 5691   Fn wfn 6558  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571
This theorem is referenced by:  fnelnfp  7197  fnnfpeq0  7198  f1omvdcnv  19477  pmtrmvd  19489  pmtrdifellem4  19512  sygbasnfpfi  19545
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