Theorem fndifnfp 7175

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 6718	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
2		fssxp 6744	. . . . . . . 8 ⊢ (𝐹:𝐴⟶V → 𝐹 ⊆ (𝐴 × V))
3	1, 2	sylbi 216	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → 𝐹 ⊆ (𝐴 × V))
4		ssdif0 4362	. . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × V) ↔ (𝐹 ∖ (𝐴 × V)) = ∅)
5	3, 4	sylib 217	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ (𝐴 × V)) = ∅)
6	5	uneq2d 4162	. . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ ∅))
7		un0 4389	. . . . 5 ⊢ ((𝐹 ∖ I ) ∪ ∅) = (𝐹 ∖ I )
8	6, 7	eqtr2di 2787	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))))
9		df-res 5687	. . . . . 6 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
10	9	difeq2i 4118	. . . . 5 ⊢ (𝐹 ∖ ( I ↾ 𝐴)) = (𝐹 ∖ ( I ∩ (𝐴 × V)))
11		difindi 4280	. . . . 5 ⊢ (𝐹 ∖ ( I ∩ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
12	10, 11	eqtri 2758	. . . 4 ⊢ (𝐹 ∖ ( I ↾ 𝐴)) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))
13	8, 12	eqtr4di 2788	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = (𝐹 ∖ ( I ↾ 𝐴)))
14	13	dmeqd 5904	. 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ ( I ↾ 𝐴)))
15		fnresi 6678	. . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴
16		fndmdif 7042	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
17	15, 16	mpan2 687	. 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥)})
18		fvresi 7172	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
19	18	neeq2d 2999	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) ≠ 𝑥))
20	19	rabbiia 3434	. . 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 2774	1 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  {crab 3430  Vcvv 3472   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321   I cid 5572   × cxp 5673  dom cdm 5675   ↾ cres 5677   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550

This theorem is referenced by:  fnelnfp  7176  fnnfpeq0  7177  f1omvdcnv  19353  pmtrmvd  19365  pmtrdifellem4  19388  sygbasnfpfi  19421