Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvmpo2 | Structured version Visualization version GIF version |
Description: Rule to change the second bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
cbvmpo2.1 | ⊢ Ⅎ𝑦𝐴 |
cbvmpo2.2 | ⊢ Ⅎ𝑤𝐴 |
cbvmpo2.3 | ⊢ Ⅎ𝑤𝐶 |
cbvmpo2.4 | ⊢ Ⅎ𝑦𝐸 |
cbvmpo2.5 | ⊢ (𝑦 = 𝑤 → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
cbvmpo2 | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvmpo2.2 | . . . . . 6 ⊢ Ⅎ𝑤𝐴 | |
2 | 1 | nfcri 2894 | . . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴 |
3 | nfcv 2907 | . . . . . 6 ⊢ Ⅎ𝑤𝐵 | |
4 | 3 | nfcri 2894 | . . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵 |
5 | 2, 4 | nfan 1907 | . . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) |
6 | cbvmpo2.3 | . . . . 5 ⊢ Ⅎ𝑤𝐶 | |
7 | 6 | nfeq2 2924 | . . . 4 ⊢ Ⅎ𝑤 𝑢 = 𝐶 |
8 | 5, 7 | nfan 1907 | . . 3 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) |
9 | cbvmpo2.1 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
10 | 9 | nfcri 2894 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
11 | nfv 1922 | . . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐵 | |
12 | 10, 11 | nfan 1907 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) |
13 | cbvmpo2.4 | . . . . 5 ⊢ Ⅎ𝑦𝐸 | |
14 | 13 | nfeq2 2924 | . . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸 |
15 | 12, 14 | nfan 1907 | . . 3 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸) |
16 | eleq1w 2822 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵)) | |
17 | 16 | anbi2d 632 | . . . 4 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) |
18 | cbvmpo2.5 | . . . . 5 ⊢ (𝑦 = 𝑤 → 𝐶 = 𝐸) | |
19 | 18 | eqeq2d 2750 | . . . 4 ⊢ (𝑦 = 𝑤 → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸)) |
20 | 17, 19 | anbi12d 634 | . . 3 ⊢ (𝑦 = 𝑤 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸))) |
21 | 8, 15, 20 | cbvoprab2 7320 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {〈〈𝑥, 𝑤〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)} |
22 | df-mpo 7239 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} | |
23 | df-mpo 7239 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸) = {〈〈𝑥, 𝑤〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)} | |
24 | 21, 22, 23 | 3eqtr4i 2777 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2112 Ⅎwnfc 2887 {coprab 7235 ∈ cmpo 7236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-nul 4254 df-if 4456 df-sn 4558 df-pr 4560 df-op 4564 df-oprab 7238 df-mpo 7239 |
This theorem is referenced by: smflimlem4 44026 |
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