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Theorem fndifnfp 6709
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 6293 . . . . . . . 8 (𝐹 Fn 𝐴𝐹:𝐴⟶V)
2 fssxp 6310 . . . . . . . 8 (𝐹:𝐴⟶V → 𝐹 ⊆ (𝐴 × V))
31, 2sylbi 209 . . . . . . 7 (𝐹 Fn 𝐴𝐹 ⊆ (𝐴 × V))
4 ssdif0 4172 . . . . . . 7 (𝐹 ⊆ (𝐴 × V) ↔ (𝐹 ∖ (𝐴 × V)) = ∅)
53, 4sylib 210 . . . . . 6 (𝐹 Fn 𝐴 → (𝐹 ∖ (𝐴 × V)) = ∅)
65uneq2d 3990 . . . . 5 (𝐹 Fn 𝐴 → ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ ∅))
7 un0 4193 . . . . 5 ((𝐹 ∖ I ) ∪ ∅) = (𝐹 ∖ I )
86, 7syl6req 2831 . . . 4 (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))))
9 df-res 5367 . . . . . 6 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
109difeq2i 3948 . . . . 5 (𝐹 ∖ ( I ↾ 𝐴)) = (𝐹 ∖ ( I ∩ (𝐴 × V)))
11 difindi 4108 . . . . 5 (𝐹 ∖ ( I ∩ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
1210, 11eqtri 2802 . . . 4 (𝐹 ∖ ( I ↾ 𝐴)) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
138, 12syl6eqr 2832 . . 3 (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = (𝐹 ∖ ( I ↾ 𝐴)))
1413dmeqd 5571 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ ( I ↾ 𝐴)))
15 fnresi 6254 . . 3 ( I ↾ 𝐴) Fn 𝐴
16 fndmdif 6584 . . 3 ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
1715, 16mpan2 681 . 2 (𝐹 Fn 𝐴 → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
18 fvresi 6706 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
1918neeq2d 3029 . . . 4 (𝑥𝐴 → ((𝐹𝑥) ≠ (( I ↾ 𝐴)‘𝑥) ↔ (𝐹𝑥) ≠ 𝑥))
2019rabbiia 3381 . . 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 2818 1 (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  wne 2969  {crab 3094  Vcvv 3398  cdif 3789  cun 3790  cin 3791  wss 3792  c0 4141   I cid 5260   × cxp 5353  dom cdm 5355  cres 5357   Fn wfn 6130  wf 6131  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143
This theorem is referenced by:  fnelnfp  6710  fnnfpeq0  6711  f1omvdcnv  18247  pmtrmvd  18259  pmtrdifellem4  18282  sygbasnfpfi  18316
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