![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > indpi1 | Structured version Visualization version GIF version |
Description: Preimage of the singleton {1} by the indicator function. See i1f1lem 25738. (Contributed by Thierry Arnoux, 21-Aug-2017.) |
Ref | Expression |
---|---|
indpi1 | ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (◡((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ind1a 34000 | . . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑥 ∈ 𝑂) → ((((𝟭‘𝑂)‘𝐴)‘𝑥) = 1 ↔ 𝑥 ∈ 𝐴)) | |
2 | 1 | 3expia 1120 | . . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (𝑥 ∈ 𝑂 → ((((𝟭‘𝑂)‘𝐴)‘𝑥) = 1 ↔ 𝑥 ∈ 𝐴))) |
3 | 2 | pm5.32d 577 | . . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝑥 ∈ 𝑂 ∧ (((𝟭‘𝑂)‘𝐴)‘𝑥) = 1) ↔ (𝑥 ∈ 𝑂 ∧ 𝑥 ∈ 𝐴))) |
4 | indf 33996 | . . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1}) | |
5 | ffn 6737 | . . . 4 ⊢ (((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1} → ((𝟭‘𝑂)‘𝐴) Fn 𝑂) | |
6 | fniniseg 7080 | . . . 4 ⊢ (((𝟭‘𝑂)‘𝐴) Fn 𝑂 → (𝑥 ∈ (◡((𝟭‘𝑂)‘𝐴) “ {1}) ↔ (𝑥 ∈ 𝑂 ∧ (((𝟭‘𝑂)‘𝐴)‘𝑥) = 1))) | |
7 | 4, 5, 6 | 3syl 18 | . . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (𝑥 ∈ (◡((𝟭‘𝑂)‘𝐴) “ {1}) ↔ (𝑥 ∈ 𝑂 ∧ (((𝟭‘𝑂)‘𝐴)‘𝑥) = 1))) |
8 | ssel 3989 | . . . . 5 ⊢ (𝐴 ⊆ 𝑂 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑂)) | |
9 | 8 | pm4.71rd 562 | . . . 4 ⊢ (𝐴 ⊆ 𝑂 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝑂 ∧ 𝑥 ∈ 𝐴))) |
10 | 9 | adantl 481 | . . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝑂 ∧ 𝑥 ∈ 𝐴))) |
11 | 3, 7, 10 | 3bitr4d 311 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (𝑥 ∈ (◡((𝟭‘𝑂)‘𝐴) “ {1}) ↔ 𝑥 ∈ 𝐴)) |
12 | 11 | eqrdv 2733 | 1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (◡((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 {cpr 4633 ◡ccnv 5688 “ cima 5692 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 0cc0 11153 1c1 11154 𝟭cind 33991 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-i2m1 11221 ax-1ne0 11222 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 |
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-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 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-uni 4913 df-iun 4998 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-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-ind 33992 |
This theorem is referenced by: indf1ofs 34007 eulerpartlemgf 34361 |
Copyright terms: Public domain | W3C validator |