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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 4077 | . . . . 5 ⊢ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ⊆ 𝐽 | |
2 | elpw2g 5350 | . . . . 5 ⊢ (𝐽 ∈ 𝑉 → ({𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ∈ 𝒫 𝐽 ↔ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ⊆ 𝐽)) | |
3 | 1, 2 | mpbiri 257 | . . . 4 ⊢ (𝐽 ∈ 𝑉 → {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ∈ 𝒫 𝐽) |
4 | 3 | adantr 479 | . . 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 7129 | . 2 ⊢ (𝐽 ∈ 𝑉 → 𝐹:𝑋⟶𝒫 𝐽) |
7 | 6 | ffnd 6728 | 1 ⊢ (𝐽 ∈ 𝑉 → 𝐹 Fn 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {crab 3430 ⊆ wss 3949 𝒫 cpw 4606 ↦ cmpt 5235 Fn wfn 6548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-fun 6555 df-fn 6556 df-f 6557 |
This theorem is referenced by: kqtopon 23651 kqid 23652 ist0-4 23653 kqfvima 23654 kqsat 23655 kqdisj 23656 kqcldsat 23657 kqopn 23658 kqcld 23659 kqt0lem 23660 isr0 23661 r0cld 23662 regr1lem2 23664 kqreglem1 23665 kqreglem2 23666 kqnrmlem1 23667 kqnrmlem2 23668 |
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