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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 2897 | . . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴 |
| 3 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑤𝐵 | |
| 4 | 3 | nfcri 2897 | . . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵 |
| 5 | 2, 4 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) |
| 6 | cbvmpo2.3 | . . . . 5 ⊢ Ⅎ𝑤𝐶 | |
| 7 | 6 | nfeq2 2923 | . . . 4 ⊢ Ⅎ𝑤 𝑢 = 𝐶 |
| 8 | 5, 7 | nfan 1899 | . . 3 ⊢ Ⅎ𝑤((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) |
| 9 | cbvmpo2.1 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
| 10 | 9 | nfcri 2897 | . . . . 5 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
| 11 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐵 | |
| 12 | 10, 11 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) |
| 13 | cbvmpo2.4 | . . . . 5 ⊢ Ⅎ𝑦𝐸 | |
| 14 | 13 | nfeq2 2923 | . . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸 |
| 15 | 12, 14 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸) |
| 16 | eleq1w 2824 | . . . . 5 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵)) | |
| 17 | 16 | anbi2d 630 | . . . 4 ⊢ (𝑦 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) |
| 18 | cbvmpo2.5 | . . . . 5 ⊢ (𝑦 = 𝑤 → 𝐶 = 𝐸) | |
| 19 | 18 | eqeq2d 2748 | . . . 4 ⊢ (𝑦 = 𝑤 → (𝑢 = 𝐶 ↔ 𝑢 = 𝐸)) |
| 20 | 17, 19 | anbi12d 632 | . . 3 ⊢ (𝑦 = 𝑤 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸))) |
| 21 | 8, 15, 20 | cbvoprab2 7521 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {〈〈𝑥, 𝑤〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)} |
| 22 | df-mpo 7436 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} | |
| 23 | df-mpo 7436 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸) = {〈〈𝑥, 𝑤〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)} | |
| 24 | 21, 22, 23 | 3eqtr4i 2775 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 Ⅎwnfc 2890 {coprab 7432 ∈ cmpo 7433 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: smflimlem4 46789 |
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