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 2970 | . . . . 5 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴 |
3 | nfcv 2976 | . . . . . 6 ⊢ Ⅎ𝑤𝐵 | |
4 | 3 | nfcri 2970 | . . . . 5 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐵 |
5 | 2, 4 | nfan 1899 | . . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) |
6 | cbvmpo2.3 | . . . . 5 ⊢ Ⅎ𝑤𝐶 | |
7 | 6 | nfeq2 2994 | . . . 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 2970 | . . . . 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 2994 | . . . 4 ⊢ Ⅎ𝑦 𝑢 = 𝐸 |
15 | 12, 14 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸) |
16 | eleq1w 2894 | . . . . 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 2831 | . . . 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 7235 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} = {〈〈𝑥, 𝑤〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)} |
22 | df-mpo 7154 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 = 𝐶)} | |
23 | df-mpo 7154 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸) = {〈〈𝑥, 𝑤〉, 𝑢〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑢 = 𝐸)} | |
24 | 21, 22, 23 | 3eqtr4i 2853 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Ⅎwnfc 2960 {coprab 7150 ∈ cmpo 7151 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-oprab 7153 df-mpo 7154 |
This theorem is referenced by: smflimlem4 43124 |
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