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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mptssid | Structured version Visualization version GIF version |
Description: The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
mptssid.1 | ⊢ Ⅎ𝑥𝐴 |
mptssid.2 | ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
Ref | Expression |
---|---|
mptssid | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvisset 3492 | . . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V) | |
2 | 1 | anim2i 618 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V)) |
3 | rabid 3453 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V)) | |
4 | 2, 3 | sylibr 233 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}) |
5 | mptssid.2 | . . . . . 6 ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} | |
6 | 4, 5 | eleqtrrdi 2845 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐶) |
7 | simpr 486 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) | |
8 | 6, 7 | jca 513 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)) |
9 | mptssid.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
10 | 9 | ssrab2f 43806 | . . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴 |
11 | 5, 10 | eqsstri 4017 | . . . . . 6 ⊢ 𝐶 ⊆ 𝐴 |
12 | 11 | sseli 3979 | . . . . 5 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) |
13 | 12 | anim1i 616 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) |
14 | 8, 13 | impbii 208 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)) |
15 | 14 | opabbii 5216 | . 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)} |
16 | df-mpt 5233 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
17 | df-mpt 5233 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)} | |
18 | 15, 16, 17 | 3eqtr4i 2771 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 Ⅎwnfc 2884 {crab 3433 Vcvv 3475 {copab 5211 ↦ cmpt 5232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rab 3434 df-v 3477 df-in 3956 df-ss 3966 df-opab 5212 df-mpt 5233 |
This theorem is referenced by: limsupequzmpt2 44434 liminfequzmpt2 44507 |
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