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Mirrors > Home > MPE Home > Th. List > fndifnfp | Structured version Visualization version GIF version |
Description: Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
Ref | Expression |
---|---|
fndifnfp | ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn2 6729 | . . . . . . . 8 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) | |
2 | fssxp 6756 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶V → 𝐹 ⊆ (𝐴 × V)) | |
3 | 1, 2 | sylbi 216 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → 𝐹 ⊆ (𝐴 × V)) |
4 | ssdif0 4367 | . . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × V) ↔ (𝐹 ∖ (𝐴 × V)) = ∅) | |
5 | 3, 4 | sylib 217 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ (𝐴 × V)) = ∅) |
6 | 5 | uneq2d 4164 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ ∅)) |
7 | un0 4394 | . . . . 5 ⊢ ((𝐹 ∖ I ) ∪ ∅) = (𝐹 ∖ I ) | |
8 | 6, 7 | eqtr2di 2785 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))) |
9 | df-res 5694 | . . . . . 6 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V)) | |
10 | 9 | difeq2i 4119 | . . . . 5 ⊢ (𝐹 ∖ ( I ↾ 𝐴)) = (𝐹 ∖ ( I ∩ (𝐴 × V))) |
11 | difindi 4284 | . . . . 5 ⊢ (𝐹 ∖ ( I ∩ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) | |
12 | 10, 11 | eqtri 2756 | . . . 4 ⊢ (𝐹 ∖ ( I ↾ 𝐴)) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) |
13 | 8, 12 | eqtr4di 2786 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = (𝐹 ∖ ( I ↾ 𝐴))) |
14 | 13 | dmeqd 5912 | . 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ ( I ↾ 𝐴))) |
15 | fnresi 6689 | . . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
16 | fndmdif 7056 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥)}) | |
17 | 15, 16 | mpan2 689 | . 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥)}) |
18 | fvresi 7188 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
19 | 18 | neeq2d 2998 | . . . 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 2772 | 1 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 {crab 3430 Vcvv 3473 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4326 I cid 5579 × cxp 5680 dom cdm 5682 ↾ cres 5684 Fn wfn 6548 ⟶wf 6549 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 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 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 |
This theorem is referenced by: fnelnfp 7192 fnnfpeq0 7193 f1omvdcnv 19413 pmtrmvd 19425 pmtrdifellem4 19448 sygbasnfpfi 19481 |
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