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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 | inres 5874 | . . . . . 6 ⊢ ( I ∩ (𝐹 ↾ 𝐴)) = (( I ∩ 𝐹) ↾ 𝐴) | |
2 | incom 4181 | . . . . . . 7 ⊢ ( I ∩ 𝐹) = (𝐹 ∩ I ) | |
3 | 2 | reseq1i 5852 | . . . . . 6 ⊢ (( I ∩ 𝐹) ↾ 𝐴) = ((𝐹 ∩ I ) ↾ 𝐴) |
4 | 1, 3 | eqtri 2847 | . . . . 5 ⊢ ( I ∩ (𝐹 ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴) |
5 | incom 4181 | . . . . 5 ⊢ ((𝐹 ↾ 𝐴) ∩ I ) = ( I ∩ (𝐹 ↾ 𝐴)) | |
6 | inres 5874 | . . . . 5 ⊢ (𝐹 ∩ ( I ↾ 𝐴)) = ((𝐹 ∩ I ) ↾ 𝐴) | |
7 | 4, 5, 6 | 3eqtr4i 2857 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴)) |
8 | fnresdm 6469 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) | |
9 | 8 | ineq1d 4191 | . . . 4 ⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐴) ∩ I ) = (𝐹 ∩ I )) |
10 | 7, 9 | syl5reqr 2874 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∩ I ) = (𝐹 ∩ ( I ↾ 𝐴))) |
11 | 10 | dmeqd 5777 | . 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = dom (𝐹 ∩ ( I ↾ 𝐴))) |
12 | fnresi 6479 | . . 3 ⊢ ( I ↾ 𝐴) Fn 𝐴 | |
13 | fndmin 6818 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ( I ↾ 𝐴) Fn 𝐴) → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)}) | |
14 | 12, 13 | mpan2 689 | . 2 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ ( I ↾ 𝐴)) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥)}) |
15 | fvresi 6938 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
16 | 15 | eqeq2d 2835 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹‘𝑥) = (( I ↾ 𝐴)‘𝑥) ↔ (𝐹‘𝑥) = 𝑥)) |
17 | 16 | rabbiia 3475 | . . 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 2863 | 1 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 {crab 3145 ∩ cin 3938 I cid 5462 dom cdm 5558 ↾ cres 5560 Fn wfn 6353 ‘cfv 6358 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-res 5570 df-iota 6317 df-fun 6360 df-fn 6361 df-fv 6366 |
This theorem is referenced by: fnelfp 6940 |
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