|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > kqffn | Structured version Visualization version GIF version | ||
| Description: The topological indistinguishability map is a function on the base. (Contributed by Mario Carneiro, 25-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| kqval.2 | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | 
| Ref | Expression | 
|---|---|
| kqffn | ⊢ (𝐽 ∈ 𝑉 → 𝐹 Fn 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssrab2 4079 | . . . . 5 ⊢ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ⊆ 𝐽 | |
| 2 | elpw2g 5332 | . . . . 5 ⊢ (𝐽 ∈ 𝑉 → ({𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ∈ 𝒫 𝐽 ↔ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ⊆ 𝐽)) | |
| 3 | 1, 2 | mpbiri 258 | . . . 4 ⊢ (𝐽 ∈ 𝑉 → {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ∈ 𝒫 𝐽) | 
| 4 | 3 | adantr 480 | . . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑥 ∈ 𝑋) → {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ∈ 𝒫 𝐽) | 
| 5 | kqval.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) | |
| 6 | 4, 5 | fmptd 7133 | . 2 ⊢ (𝐽 ∈ 𝑉 → 𝐹:𝑋⟶𝒫 𝐽) | 
| 7 | 6 | ffnd 6736 | 1 ⊢ (𝐽 ∈ 𝑉 → 𝐹 Fn 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 {crab 3435 ⊆ wss 3950 𝒫 cpw 4599 ↦ cmpt 5224 Fn wfn 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: kqtopon 23736 kqid 23737 ist0-4 23738 kqfvima 23739 kqsat 23740 kqdisj 23741 kqcldsat 23742 kqopn 23743 kqcld 23744 kqt0lem 23745 isr0 23746 r0cld 23747 regr1lem2 23749 kqreglem1 23750 kqreglem2 23751 kqnrmlem1 23752 kqnrmlem2 23753 | 
| Copyright terms: Public domain | W3C validator |