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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 4090 | . . . . 5 ⊢ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ⊆ 𝐽 | |
2 | elpw2g 5339 | . . . . 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 7134 | . 2 ⊢ (𝐽 ∈ 𝑉 → 𝐹:𝑋⟶𝒫 𝐽) |
7 | 6 | ffnd 6738 | 1 ⊢ (𝐽 ∈ 𝑉 → 𝐹 Fn 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 ⊆ wss 3963 𝒫 cpw 4605 ↦ cmpt 5231 Fn wfn 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: kqtopon 23751 kqid 23752 ist0-4 23753 kqfvima 23754 kqsat 23755 kqdisj 23756 kqcldsat 23757 kqopn 23758 kqcld 23759 kqt0lem 23760 isr0 23761 r0cld 23762 regr1lem2 23764 kqreglem1 23765 kqreglem2 23766 kqnrmlem1 23767 kqnrmlem2 23768 |
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