Theorem fndifnfp 6915

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 6489	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
2		fssxp 6508	. . . . . . . 8 ⊢ (𝐹:𝐴⟶V → 𝐹 ⊆ (𝐴 × V))
3	1, 2	sylbi 220	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → 𝐹 ⊆ (𝐴 × V))
4		ssdif0 4277	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × V) ↔ (𝐹 ∖ (𝐴 × V)) = ∅)
5	3, 4	sylib 221	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ (𝐴 × V)) = ∅)
6	5	uneq2d 4090	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ ∅))
7		un0 4298	. . . . 5 ⊢ ((𝐹 ∖ I ) ∪ ∅) = (𝐹 ∖ I )
8	6, 7	eqtr2di 2850	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))))
9		df-res 5531	. . . . . 6 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
10	9	difeq2i 4047	. . . . 5 ⊢ (𝐹 ∖ ( I ↾ 𝐴)) = (𝐹 ∖ ( I ∩ (𝐴 × V)))
11		difindi 4208	. . . . 5 ⊢ (𝐹 ∖ ( I ∩ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
12	10, 11	eqtri 2821	. . . 4 ⊢ (𝐹 ∖ ( I ↾ 𝐴)) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
13	8, 12	eqtr4di 2851	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = (𝐹 ∖ ( I ↾ 𝐴)))
14	13	dmeqd 5738	. 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ ( I ↾ 𝐴)))
15		fnresi 6448	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
16		fndmdif 6789	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
17	15, 16	mpan2 690	. 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
18		fvresi 6912	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
19	18	neeq2d 3047	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) ≠ 𝑥))
20	19	rabbiia 3419	. . 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 2837	1 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243   I cid 5424   × cxp 5517  dom cdm 5519   ↾ cres 5521   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332

This theorem is referenced by:  fnelnfp  6916  fnnfpeq0  6917  f1omvdcnv  18564  pmtrmvd  18576  pmtrdifellem4  18599  sygbasnfpfi  18632