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Theorem fndifnfp 7190
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 6730 . . . . . . . 8 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
2 fssxp 6756 . . . . . . . 8 (𝐹:𝐴⟶V → 𝐹 ⊆ (𝐴 × V))
31, 2sylbi 216 . . . . . . 7 (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × V))
4 ssdif0 4366 . . . . . . 7 (𝐹 ⊆ (𝐴 × V) ↔ (𝐹 ∖ (𝐴 × V)) = ∅)
53, 4sylib 217 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹 ∖ (𝐴 × V)) = ∅)
65uneq2d 4163 . . . . 5 (𝐹 Fn 𝐴 → ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ ∅))
7 un0 4395 . . . . 5 ((𝐹 ∖ I ) ∪ ∅) = (𝐹 ∖ I )
86, 7eqtr2di 2783 . . . 4 (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))))
9 df-res 5694 . . . . . 6 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
109difeq2i 4118 . . . . 5 (𝐹 ∖ ( I ↾ 𝐴)) = (𝐹 ∖ ( I ∩ (𝐴 × V)))
11 difindi 4283 . . . . 5 (𝐹 ∖ ( I ∩ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
1210, 11eqtri 2754 . . . 4 (𝐹 ∖ ( I ↾ 𝐴)) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
138, 12eqtr4di 2784 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = (𝐹 ∖ ( I ↾ 𝐴)))
1413dmeqd 5912 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ ( I ↾ 𝐴)))
15 fnresi 6690 . . 3 ( I ↾ 𝐴) Fn 𝐴
16 fndmdif 7055 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
1715, 16mpan2 689 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
18 fvresi 7187 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
1918neeq2d 2991 . . . 4 (𝑥𝐴 → ((𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) ≠ 𝑥))
2019rabbiia 3423 . . 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 2770 1 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wne 2930  {crab 3419  Vcvv 3462  cdif 3944  cun 3945  cin 3946  wss 3947  c0 4325   I cid 5579   × cxp 5680  dom cdm 5682  cres 5684   Fn wfn 6549  wf 6550  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562
This theorem is referenced by:  fnelnfp  7191  fnnfpeq0  7192  f1omvdcnv  19442  pmtrmvd  19454  pmtrdifellem4  19477  sygbasnfpfi  19510
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