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Mirrors > Home > MPE Home > Th. List > mrcflem | Structured version Visualization version GIF version |
Description: The domain and codomain 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 482 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝐶 ∈ (Moore‘𝑋)) | |
2 | ssrab2 4089 | . . . 4 ⊢ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶 | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶) |
4 | sseq2 4021 | . . . . 5 ⊢ (𝑠 = 𝑋 → (𝑥 ⊆ 𝑠 ↔ 𝑥 ⊆ 𝑋)) | |
5 | mre1cl 17638 | . . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ 𝐶) |
7 | elpwi 4611 | . . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
8 | 7 | adantl 481 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑥 ⊆ 𝑋) |
9 | 4, 6, 8 | elrabd 3696 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → 𝑋 ∈ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠}) |
10 | 9 | ne0d 4347 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ≠ ∅) |
11 | mreintcl 17639 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ⊆ 𝐶 ∧ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ≠ ∅) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ∈ 𝐶) | |
12 | 1, 3, 10, 11 | syl3anc 1370 | . 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑥 ∈ 𝒫 𝑋) → ∩ {𝑠 ∈ 𝐶 ∣ 𝑥 ⊆ 𝑠} ∈ 𝐶) |
13 | 12 | fmpttd 7134 | 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 395 ∈ wcel 2105 ≠ wne 2937 {crab 3432 ⊆ wss 3962 ∅c0 4338 𝒫 cpw 4604 ∩ cint 4950 ↦ cmpt 5230 ⟶wf 6558 ‘cfv 6562 Moorecmre 17626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-mre 17630 |
This theorem is referenced by: fnmrc 17651 mrcfval 17652 mrcf 17653 |
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