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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvmpo1 | Structured version Visualization version GIF version |
Description: Rule to change the first bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
cbvmpo1.1 | ⊢ Ⅎ𝑥𝐵 |
cbvmpo1.2 | ⊢ Ⅎ𝑧𝐵 |
cbvmpo1.3 | ⊢ Ⅎ𝑧𝐶 |
cbvmpo1.4 | ⊢ Ⅎ𝑥𝐸 |
cbvmpo1.5 | ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸) |
Ref | Expression |
---|---|
cbvmpo1 | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴 | |
2 | cbvmpo1.2 | . . . . . 6 ⊢ Ⅎ𝑧𝐵 | |
3 | 2 | nfcri 2890 | . . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵 |
4 | 1, 3 | nfan 1902 | . . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) |
5 | cbvmpo1.3 | . . . . 5 ⊢ Ⅎ𝑧𝐶 | |
6 | 5 | nfeq2 2920 | . . . 4 ⊢ Ⅎ𝑧 𝑢 = 𝐶 |
7 | 4, 6 | nfan 1902 | . . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) |
8 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 | |
9 | cbvmpo1.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
10 | 9 | nfcri 2890 | . . . . 5 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
11 | 8, 10 | nfan 1902 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) |
12 | cbvmpo1.4 | . . . . 5 ⊢ Ⅎ𝑥𝐸 | |
13 | 12 | nfeq2 2920 | . . . 4 ⊢ Ⅎ𝑥 𝑢 = 𝐸 |
14 | 11, 13 | nfan 1902 | . . 3 ⊢ Ⅎ𝑥((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸) |
15 | eleq1w 2816 | . . . . 5 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) | |
16 | 15 | anbi1d 630 | . . . 4 ⊢ (𝑥 = 𝑧 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
17 | cbvmpo1.5 | . . . . 5 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸) | |
18 | 17 | eqeq2d 2743 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸)) |
19 | 16, 18 | anbi12d 631 | . . 3 ⊢ (𝑥 = 𝑧 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸))) |
20 | 7, 14, 19 | cbvoprab1 7495 | . 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸)} |
21 | df-mpo 7413 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} | |
22 | df-mpo 7413 | . 2 ⊢ (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐸)} | |
23 | 20, 21, 22 | 3eqtr4i 2770 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 Ⅎwnfc 2883 {coprab 7409 ∈ cmpo 7410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-opab 5211 df-oprab 7412 df-mpo 7413 |
This theorem is referenced by: (None) |
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