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Mirrors > Home > ILE Home > Th. List > cbvmpo | GIF version |
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.) |
Ref | Expression |
---|---|
cbvmpo.1 | ⊢ Ⅎ𝑧𝐶 |
cbvmpo.2 | ⊢ Ⅎ𝑤𝐶 |
cbvmpo.3 | ⊢ Ⅎ𝑥𝐷 |
cbvmpo.4 | ⊢ Ⅎ𝑦𝐷 |
cbvmpo.5 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
cbvmpo | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2306 | . 2 ⊢ Ⅎ𝑧𝐵 | |
2 | nfcv 2306 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | cbvmpo.1 | . 2 ⊢ Ⅎ𝑧𝐶 | |
4 | cbvmpo.2 | . 2 ⊢ Ⅎ𝑤𝐶 | |
5 | cbvmpo.3 | . 2 ⊢ Ⅎ𝑥𝐷 | |
6 | cbvmpo.4 | . 2 ⊢ Ⅎ𝑦𝐷 | |
7 | eqidd 2165 | . 2 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐵) | |
8 | cbvmpo.5 | . 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | cbvmpox 5911 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 Ⅎwnfc 2293 ∈ cmpo 5838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-oprab 5840 df-mpo 5841 |
This theorem is referenced by: cbvmpov 5913 fnmpoovd 6174 fmpoco 6175 xpf1o 6801 cnmpt2t 12840 |
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