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Mirrors > Home > ILE Home > Th. List > cbvmpt2 | 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 |
---|---|
cbvmpt2.1 | ⊢ Ⅎ𝑧𝐶 |
cbvmpt2.2 | ⊢ Ⅎ𝑤𝐶 |
cbvmpt2.3 | ⊢ Ⅎ𝑥𝐷 |
cbvmpt2.4 | ⊢ Ⅎ𝑦𝐷 |
cbvmpt2.5 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
cbvmpt2 | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2178 | . 2 ⊢ Ⅎ𝑧𝐵 | |
2 | nfcv 2178 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | cbvmpt2.1 | . 2 ⊢ Ⅎ𝑧𝐶 | |
4 | cbvmpt2.2 | . 2 ⊢ Ⅎ𝑤𝐶 | |
5 | cbvmpt2.3 | . 2 ⊢ Ⅎ𝑥𝐷 | |
6 | cbvmpt2.4 | . 2 ⊢ Ⅎ𝑦𝐷 | |
7 | eqidd 2041 | . 2 ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐵) | |
8 | cbvmpt2.5 | . 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | cbvmpt2x 5582 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 Ⅎwnfc 2165 ↦ cmpt2 5514 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-oprab 5516 df-mpt2 5517 |
This theorem is referenced by: cbvmpt2v 5584 fmpt2co 5837 |
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