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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 6518 | . . . . . . . 8 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) | |
2 | fssxp 6536 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶V → 𝐹 ⊆ (𝐴 × V)) | |
3 | 1, 2 | sylbi 219 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → 𝐹 ⊆ (𝐴 × V)) |
4 | ssdif0 4325 | . . . . . . 7 ⊢ (𝐹 ⊆ (𝐴 × V) ↔ (𝐹 ∖ (𝐴 × V)) = ∅) | |
5 | 3, 4 | sylib 220 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ (𝐴 × V)) = ∅) |
6 | 5 | uneq2d 4141 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ ∅)) |
7 | un0 4346 | . . . . 5 ⊢ ((𝐹 ∖ I ) ∪ ∅) = (𝐹 ∖ I ) | |
8 | 6, 7 | syl6req 2875 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V)))) |
9 | df-res 5569 | . . . . . 6 ⊢ ( I ↾ 𝐴) = ( I ∩ (𝐴 × V)) | |
10 | 9 | difeq2i 4098 | . . . . 5 ⊢ (𝐹 ∖ ( I ↾ 𝐴)) = (𝐹 ∖ ( I ∩ (𝐴 × V))) |
11 | difindi 4260 | . . . . 5 ⊢ (𝐹 ∖ ( I ∩ (𝐴 × V))) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) | |
12 | 10, 11 | eqtri 2846 | . . . 4 ⊢ (𝐹 ∖ ( I ↾ 𝐴)) = ((𝐹 ∖ I ) ∪ (𝐹 ∖ (𝐴 × V))) |
13 | 8, 12 | syl6eqr 2876 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∖ I ) = (𝐹 ∖ ( I ↾ 𝐴))) |
14 | 13 | dmeqd 5776 | . 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = dom (𝐹 ∖ ( I ↾ 𝐴))) |
15 | fnresi 6478 | . . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
16 | fndmdif 6814 | . . 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 6937 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
19 | 18 | neeq2d 3078 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) ≠ (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) ≠ 𝑥)) |
20 | 19 | rabbiia 3474 | . . 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 2862 | 1 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {crab 3144 Vcvv 3496 ∖ cdif 3935 ∪ cun 3936 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 I cid 5461 × cxp 5555 dom cdm 5557 ↾ cres 5559 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 |
This theorem is referenced by: fnelnfp 6941 fnnfpeq0 6942 f1omvdcnv 18574 pmtrmvd 18586 pmtrdifellem4 18609 sygbasnfpfi 18642 |
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