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Theorem fndifnfp 6483
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 6085 . . . . . . . 8 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
2 fssxp 6098 . . . . . . . 8 (𝐹:𝐴⟶V → 𝐹 ⊆ (𝐴 × V))
31, 2sylbi 207 . . . . . . 7 (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × V))
4 ssdif0 3975 . . . . . . 7 (𝐹 ⊆ (𝐴 × V) ↔ (𝐹 ∖ (𝐴 × V)) = ∅)
53, 4sylib 208 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹 ∖ (𝐴 × V)) = ∅)
65uneq2d 3800 . . . . 5 (𝐹 Fn 𝐴 → ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ ∅))
7 un0 4000 . . . . 5 ((𝐹 ∖ I ) ∪ ∅) = (𝐹 ∖ I )
86, 7syl6req 2702 . . . 4 (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))))
9 df-res 5155 . . . . . 6 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
109difeq2i 3758 . . . . 5 (𝐹 ∖ ( I ↾ 𝐴)) = (𝐹 ∖ ( I ∩ (𝐴 × V)))
11 difindi 3914 . . . . 5 (𝐹 ∖ ( I ∩ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
1210, 11eqtri 2673 . . . 4 (𝐹 ∖ ( I ↾ 𝐴)) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
138, 12syl6eqr 2703 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = (𝐹 ∖ ( I ↾ 𝐴)))
1413dmeqd 5358 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ ( I ↾ 𝐴)))
15 fnresi 6046 . . 3 ( I ↾ 𝐴) Fn 𝐴
16 fndmdif 6361 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
1715, 16mpan2 707 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
18 fvresi 6480 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
1918neeq2d 2883 . . . 4 (𝑥𝐴 → ((𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) ≠ 𝑥))
2019rabbiia 3215 . . 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 2689 1 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wne 2823  {crab 2945  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948   I cid 5052   × cxp 5141  dom cdm 5143  cres 5145   Fn wfn 5921  wf 5922  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934
This theorem is referenced by:  fnelnfp  6484  fnnfpeq0  6485  f1omvdcnv  17910  pmtrmvd  17922  pmtrdifellem4  17945  sygbasnfpfi  17978
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