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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 3914 | . . . . 5 ⊢ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ⊆ 𝐽 | |
2 | elpw2g 5051 | . . . . 5 ⊢ (𝐽 ∈ 𝑉 → ({𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ∈ 𝒫 𝐽 ↔ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ⊆ 𝐽)) | |
3 | 1, 2 | mpbiri 250 | . . . 4 ⊢ (𝐽 ∈ 𝑉 → {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦} ∈ 𝒫 𝐽) |
4 | 3 | adantr 474 | . . 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 6638 | . 2 ⊢ (𝐽 ∈ 𝑉 → 𝐹:𝑋⟶𝒫 𝐽) |
7 | 6 | ffnd 6283 | 1 ⊢ (𝐽 ∈ 𝑉 → 𝐹 Fn 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 {crab 3121 ⊆ wss 3798 𝒫 cpw 4380 ↦ cmpt 4954 Fn wfn 6122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 |
This theorem is referenced by: kqtopon 21908 kqid 21909 ist0-4 21910 kqfvima 21911 kqsat 21912 kqdisj 21913 kqcldsat 21914 kqopn 21915 kqcld 21916 kqt0lem 21917 isr0 21918 r0cld 21919 regr1lem2 21921 kqreglem1 21922 kqreglem2 21923 kqnrmlem1 21924 kqnrmlem2 21925 |
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