Mathbox for Thierry Arnoux |
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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 23262. (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 29410 | . . . . 5 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂 ∧ 𝑥 ∈ 𝑂) → ((((𝟭‘𝑂)‘𝐴)‘𝑥) = 1 ↔ 𝑥 ∈ 𝐴)) | |
2 | 1 | 3expia 1259 | . . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (𝑥 ∈ 𝑂 → ((((𝟭‘𝑂)‘𝐴)‘𝑥) = 1 ↔ 𝑥 ∈ 𝐴))) |
3 | 2 | pm5.32d 669 | . . 3 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝑥 ∈ 𝑂 ∧ (((𝟭‘𝑂)‘𝐴)‘𝑥) = 1) ↔ (𝑥 ∈ 𝑂 ∧ 𝑥 ∈ 𝐴))) |
4 | indf 29405 | . . . 4 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1}) | |
5 | ffn 5958 | . . . 4 ⊢ (((𝟭‘𝑂)‘𝐴):𝑂⟶{0, 1} → ((𝟭‘𝑂)‘𝐴) Fn 𝑂) | |
6 | fniniseg 6246 | . . . 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 3562 | . . . . 5 ⊢ (𝐴 ⊆ 𝑂 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝑂)) | |
9 | 8 | pm4.71rd 665 | . . . 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 299 | . 2 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (𝑥 ∈ (◡((𝟭‘𝑂)‘𝐴) “ {1}) ↔ 𝑥 ∈ 𝐴)) |
12 | 11 | eqrdv 2608 | 1 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑂) → (◡((𝟭‘𝑂)‘𝐴) “ {1}) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 {csn 4125 {cpr 4127 ◡ccnv 5037 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 0cc0 9815 1c1 9816 𝟭cind 29400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-ind 29401 |
This theorem is referenced by: indf1ofs 29415 eulerpartlemgf 29768 |
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