Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mrcflem | Structured version Visualization version GIF version |
Description: The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.) |
Ref | Expression |
---|---|
mrcflem | ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}):𝒫 𝑋⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋)) | |
2 | ssrab2 4056 | . . . 4 ⊢ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶 | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶) |
4 | sseq2 3993 | . . . . 5 ⊢ (𝑠 = 𝑋 → (𝑥 ⊆ 𝑠 ↔ 𝑥 ⊆ 𝑋)) | |
5 | mre1cl 16865 | . . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
6 | 5 | adantr 483 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ 𝐶) |
7 | elpwi 4548 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
8 | 7 | adantl 484 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 ⊆ 𝑋) |
9 | 4, 6, 8 | elrabd 3682 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}) |
10 | 9 | ne0d 4301 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ≠ ∅) |
11 | mreintcl 16866 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶 ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ≠ ∅) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ∈ 𝐶) | |
12 | 1, 3, 10, 11 | syl3anc 1367 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ∈ 𝐶) |
13 | 12 | fmpttd 6879 | 1 ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝑥 ∈ 𝒫 𝑋 ↦ ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}):𝒫 𝑋⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ≠ wne 3016 {crab 3142 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 ∩ cint 4876 ↦ cmpt 5146 ⟶wf 6351 ‘cfv 6355 Moorecmre 16853 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-mre 16857 |
This theorem is referenced by: fnmrc 16878 mrcfval 16879 mrcf 16880 |
Copyright terms: Public domain | W3C validator |