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Mirrors > Home > MPE Home > Th. List > cbvmpt2 | Structured version Visualization version 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 2751 | . 2 ⊢ Ⅎ𝑧𝐵 | |
2 | nfcv 2751 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | cbvmpt2.1 | . 2 ⊢ Ⅎ𝑧𝐶 | |
4 | cbvmpt2.2 | . 2 ⊢ Ⅎ𝑤𝐶 | |
5 | cbvmpt2.3 | . 2 ⊢ Ⅎ𝑥𝐷 | |
6 | cbvmpt2.4 | . 2 ⊢ Ⅎ𝑦𝐷 | |
7 | eqidd 2611 | . 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 6631 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 Ⅎwnfc 2738 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-opab 4644 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: cbvmpt2v 6633 el2mpt2csbcl 7137 fnmpt2ovd 7139 fmpt2co 7147 mpt2curryd 7282 fvmpt2curryd 7284 xpf1o 8007 cnfcomlem 8479 fseqenlem1 8730 relexpsucnnr 13613 gsumdixp 18432 evlslem4 19329 madugsum 20268 cnmpt2t 21286 cnmptk2 21299 fmucnd 21906 fsum2cn 22482 fmuldfeqlem1 38649 smflim 39663 |
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