Theorem fndifnfp 7120

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 ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem fndifnfp

Step	Hyp	Ref	Expression
1		dffn2 6662	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
2		fssxp 6687	. . . . . . . 8 ⊢ (𝐹:𝐴⟶V → 𝐹 ⊆ (𝐴 × V))
3	1, 2	sylbi 217	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → 𝐹 ⊆ (𝐴 × V))
4		ssdif0 4316	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × V) ↔ (𝐹 ∖ (𝐴 × V)) = ∅)
5	3, 4	sylib 218	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ (𝐴 × V)) = ∅)
6	5	uneq2d 4118	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ ∅))
7		un0 4344	. . . . 5 ⊢ ((𝐹 ∖ I ) ∪ ∅) = (𝐹 ∖ I )
8	6, 7	eqtr2di 2786	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))))
9		df-res 5634	. . . . . 6 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
10	9	difeq2i 4073	. . . . 5 ⊢ (𝐹 ∖ ( I ↾ 𝐴)) = (𝐹 ∖ ( I ∩ (𝐴 × V)))
11		difindi 4242	. . . . 5 ⊢ (𝐹 ∖ ( I ∩ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
12	10, 11	eqtri 2757	. . . 4 ⊢ (𝐹 ∖ ( I ↾ 𝐴)) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
13	8, 12	eqtr4di 2787	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = (𝐹 ∖ ( I ↾ 𝐴)))
14	13	dmeqd 5852	. 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ ( I ↾ 𝐴)))
15		fnresi 6619	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
16		fndmdif 6985	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
17	15, 16	mpan2 691	. 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
18		fvresi 7117	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
19	18	neeq2d 2990	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) ≠ 𝑥))
20	19	rabbiia 3401	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥}
21	20	a1i 11	. 2 ⊢ (𝐹 Fn 𝐴 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})
22	14, 17, 21	3eqtrd 2773	1 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  {crab 3397  Vcvv 3438   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283   I cid 5516   × cxp 5620  dom cdm 5622   ↾ cres 5624   Fn wfn 6485  ⟶wf 6486  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498

This theorem is referenced by:  fnelnfp  7121  fnnfpeq0  7122  f1omvdcnv  19371  pmtrmvd  19383  pmtrdifellem4  19406  sygbasnfpfi  19439