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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 6738 . . . . . . . 8 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
2 fssxp 6763 . . . . . . . 8 (𝐹:𝐴⟶V → 𝐹 ⊆ (𝐴 × V))
31, 2sylbi 217 . . . . . . 7 (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × V))
4 ssdif0 4366 . . . . . . 7 (𝐹 ⊆ (𝐴 × V) ↔ (𝐹 ∖ (𝐴 × V)) = ∅)
53, 4sylib 218 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹 ∖ (𝐴 × V)) = ∅)
65uneq2d 4168 . . . . 5 (𝐹 Fn 𝐴 → ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ ∅))
7 un0 4394 . . . . 5 ((𝐹 ∖ I ) ∪ ∅) = (𝐹 ∖ I )
86, 7eqtr2di 2794 . . . 4 (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))))
9 df-res 5697 . . . . . 6 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
109difeq2i 4123 . . . . 5 (𝐹 ∖ ( I ↾ 𝐴)) = (𝐹 ∖ ( I ∩ (𝐴 × V)))
11 difindi 4292 . . . . 5 (𝐹 ∖ ( I ∩ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
1210, 11eqtri 2765 . . . 4 (𝐹 ∖ ( I ↾ 𝐴)) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
138, 12eqtr4di 2795 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = (𝐹 ∖ ( I ↾ 𝐴)))
1413dmeqd 5916 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ ( I ↾ 𝐴)))
15 fnresi 6697 . . 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 3001 . . . 4 (𝑥𝐴 → ((𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) ≠ 𝑥))
2019rabbiia 3440 . . 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 2781 1 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wne 2940  {crab 3436  Vcvv 3480  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333   I cid 5577   × cxp 5683  dom cdm 5685  cres 5687   Fn wfn 6556  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  fnelnfp  7197  fnnfpeq0  7198  f1omvdcnv  19462  pmtrmvd  19474  pmtrdifellem4  19497  sygbasnfpfi  19530
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