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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 6687 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
| 2 | 1 | ineq1d 4219 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ I ) = (𝐹 ∩ I )) |
| 3 | inres 6015 | . . . . . 6 ⊢ ( I ∩ (𝐹 ↾ 𝐴)) = (( I ∩ 𝐹) ↾ 𝐴) | |
| 4 | incom 4209 | . . . . . . 7 ⊢ ( I ∩ 𝐹) = (𝐹 ∩ I ) | |
| 5 | 4 | reseq1i 5993 | . . . . . 6 ⊢ (( I ∩ 𝐹) ↾ 𝐴) = ((𝐹 ∩ I ) ↾ 𝐴) |
| 6 | 3, 5 | eqtri 2765 | . . . . 5 ⊢ ( I ∩ (𝐹 ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴) |
| 7 | incom 4209 | . . . . 5 ⊢ ((𝐹 ↾ 𝐴) ∩ I ) = ( I ∩ (𝐹 ↾ 𝐴)) | |
| 8 | inres 6015 | . . . . 5 ⊢ (𝐹 ∩ ( I ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴) | |
| 9 | 6, 7, 8 | 3eqtr4i 2775 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴)) |
| 10 | 2, 9 | eqtr3di 2792 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴))) |
| 11 | 10 | dmeqd 5916 | . 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = dom (𝐹 ∩ ( I ↾ 𝐴))) |
| 12 | fnresi 6697 | . . 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 2748 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) = 𝑥)) |
| 17 | 16 | rabbiia 3440 | . . 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 2781 | 1 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {crab 3436 ∩ cin 3950 I cid 5577 dom cdm 5685 ↾ cres 5687 Fn wfn 6556 ‘cfv 6561 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: fnelfp 7195 |
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