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Mirrors > Home > MPE Home > Th. List > fninfp | Structured version Visualization version GIF version |
Description: Express the class of fixed points of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
fninfp | ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnresdm 6688 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
2 | 1 | ineq1d 4227 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ I ) = (𝐹 ∩ I )) |
3 | inres 6018 | . . . . . 6 ⊢ ( I ∩ (𝐹 ↾ 𝐴)) = (( I ∩ 𝐹) ↾ 𝐴) | |
4 | incom 4217 | . . . . . . 7 ⊢ ( I ∩ 𝐹) = (𝐹 ∩ I ) | |
5 | 4 | reseq1i 5996 | . . . . . 6 ⊢ (( I ∩ 𝐹) ↾ 𝐴) = ((𝐹 ∩ I ) ↾ 𝐴) |
6 | 3, 5 | eqtri 2763 | . . . . 5 ⊢ ( I ∩ (𝐹 ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴) |
7 | incom 4217 | . . . . 5 ⊢ ((𝐹 ↾ 𝐴) ∩ I ) = ( I ∩ (𝐹 ↾ 𝐴)) | |
8 | inres 6018 | . . . . 5 ⊢ (𝐹 ∩ ( I ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴) | |
9 | 6, 7, 8 | 3eqtr4i 2773 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴)) |
10 | 2, 9 | eqtr3di 2790 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴))) |
11 | 10 | dmeqd 5919 | . 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = dom (𝐹 ∩ ( I ↾ 𝐴))) |
12 | fnresi 6698 | . . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
13 | fndmin 7065 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)}) | |
14 | 12, 13 | mpan2 691 | . 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)}) |
15 | fvresi 7193 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
16 | 15 | eqeq2d 2746 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) = 𝑥)) |
17 | 16 | rabbiia 3437 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥} |
18 | 17 | a1i 11 | . 2 ⊢ (𝐹 Fn 𝐴 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}) |
19 | 11, 14, 18 | 3eqtrd 2779 | 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 2106 {crab 3433 ∩ cin 3962 I cid 5582 dom cdm 5689 ↾ cres 5691 Fn wfn 6558 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 |
This theorem is referenced by: fnelfp 7195 |
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