Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 3509 | . . . . . . . 8 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V) | |
2 | 1 | anim2i 616 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V)) |
3 | rabid 3376 | . . . . . . 7 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V)) | |
4 | 2, 3 | sylibr 235 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}) |
5 | mptssid.2 | . . . . . 6 ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} | |
6 | 4, 5 | eleqtrrdi 2921 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐶) |
7 | simpr 485 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵) | |
8 | 6, 7 | jca 512 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)) |
9 | mptssid.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
10 | 9 | ssrab2f 41260 | . . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴 |
11 | 5, 10 | eqsstri 3998 | . . . . . 6 ⊢ 𝐶 ⊆ 𝐴 |
12 | 11 | sseli 3960 | . . . . 5 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴) |
13 | 12 | anim1i 614 | . . . 4 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) |
14 | 8, 13 | impbii 210 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)) |
15 | 14 | opabbii 5124 | . 2 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)} |
16 | df-mpt 5138 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
17 | df-mpt 5138 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)} | |
18 | 15, 16, 17 | 3eqtr4i 2851 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 Ⅎwnfc 2958 {crab 3139 Vcvv 3492 {copab 5119 ↦ cmpt 5137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 df-opab 5120 df-mpt 5138 |
This theorem is referenced by: limsupequzmpt2 41875 liminfequzmpt2 41948 |
Copyright terms: Public domain | W3C validator |