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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-charfundc | GIF version |
Description: Properties of the characteristic function on the class 𝑋 of the class 𝐴, provided membership in 𝐴 is decidable in 𝑋. (Contributed by BJ, 6-Aug-2024.) |
Ref | Expression |
---|---|
bj-charfundc.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅))) |
bj-charfundc.dc | ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴) |
Ref | Expression |
---|---|
bj-charfundc | ⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-charfundc.1 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝑋 ↦ if(𝑥 ∈ 𝐴, 1o, ∅))) | |
2 | 1lt2o 6383 | . . . . 5 ⊢ 1o ∈ 2o | |
3 | 2 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 1o ∈ 2o) |
4 | 0lt2o 6382 | . . . . 5 ⊢ ∅ ∈ 2o | |
5 | 4 | a1i 9 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∅ ∈ 2o) |
6 | bj-charfundc.dc | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 DECID 𝑥 ∈ 𝐴) | |
7 | 6 | r19.21bi 2545 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → DECID 𝑥 ∈ 𝐴) |
8 | 3, 5, 7 | ifcldcd 3540 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → if(𝑥 ∈ 𝐴, 1o, ∅) ∈ 2o) |
9 | 1, 8 | fmpt3d 5620 | . 2 ⊢ (𝜑 → 𝐹:𝑋⟶2o) |
10 | inss1 3327 | . . . . . . . 8 ⊢ (𝑋 ∩ 𝐴) ⊆ 𝑋 | |
11 | 10 | a1i 9 | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∩ 𝐴) ⊆ 𝑋) |
12 | 11 | sseld 3127 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑋 ∩ 𝐴) → 𝑥 ∈ 𝑋)) |
13 | 12 | imdistani 442 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → (𝜑 ∧ 𝑥 ∈ 𝑋)) |
14 | 1, 8 | fvmpt2d 5551 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐹‘𝑥) = if(𝑥 ∈ 𝐴, 1o, ∅)) |
15 | 13, 14 | syl 14 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → (𝐹‘𝑥) = if(𝑥 ∈ 𝐴, 1o, ∅)) |
16 | simpr 109 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ (𝑋 ∩ 𝐴)) | |
17 | 16 | elin2d 3297 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → 𝑥 ∈ 𝐴) |
18 | 17 | iftrued 3512 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → if(𝑥 ∈ 𝐴, 1o, ∅) = 1o) |
19 | 15, 18 | eqtrd 2190 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∩ 𝐴)) → (𝐹‘𝑥) = 1o) |
20 | 19 | ralrimiva 2530 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o) |
21 | difssd 3234 | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∖ 𝐴) ⊆ 𝑋) | |
22 | 21 | sseld 3127 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝑋 ∖ 𝐴) → 𝑥 ∈ 𝑋)) |
23 | 22 | imdistani 442 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∖ 𝐴)) → (𝜑 ∧ 𝑥 ∈ 𝑋)) |
24 | 23, 14 | syl 14 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∖ 𝐴)) → (𝐹‘𝑥) = if(𝑥 ∈ 𝐴, 1o, ∅)) |
25 | simpr 109 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∖ 𝐴)) → 𝑥 ∈ (𝑋 ∖ 𝐴)) | |
26 | 25 | eldifbd 3114 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴) |
27 | 26 | iffalsed 3515 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∖ 𝐴)) → if(𝑥 ∈ 𝐴, 1o, ∅) = ∅) |
28 | 24, 27 | eqtrd 2190 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑋 ∖ 𝐴)) → (𝐹‘𝑥) = ∅) |
29 | 28 | ralrimiva 2530 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅) |
30 | 9, 20, 29 | jca32 308 | 1 ⊢ (𝜑 → (𝐹:𝑋⟶2o ∧ (∀𝑥 ∈ (𝑋 ∩ 𝐴)(𝐹‘𝑥) = 1o ∧ ∀𝑥 ∈ (𝑋 ∖ 𝐴)(𝐹‘𝑥) = ∅))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 820 = wceq 1335 ∈ wcel 2128 ∀wral 2435 ∖ cdif 3099 ∩ cin 3101 ⊆ wss 3102 ∅c0 3394 ifcif 3505 ↦ cmpt 4025 ⟶wf 5163 ‘cfv 5167 1oc1o 6350 2oc2o 6351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4134 ax-pr 4168 ax-un 4392 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4252 df-iord 4325 df-on 4327 df-suc 4330 df-xp 4589 df-rel 4590 df-cnv 4591 df-co 4592 df-dm 4593 df-rn 4594 df-res 4595 df-ima 4596 df-iota 5132 df-fun 5169 df-fn 5170 df-f 5171 df-fv 5175 df-1o 6357 df-2o 6358 |
This theorem is referenced by: bj-charfunbi 13346 |
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