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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 2764 | . 2 ⊢ Ⅎ𝑧𝐵 | |
2 | nfcv 2764 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | cbvmpt2.1 | . 2 ⊢ Ⅎ𝑧𝐶 | |
4 | cbvmpt2.2 | . 2 ⊢ Ⅎ𝑤𝐶 | |
5 | cbvmpt2.3 | . 2 ⊢ Ⅎ𝑥𝐷 | |
6 | cbvmpt2.4 | . 2 ⊢ Ⅎ𝑦𝐷 | |
7 | eqidd 2623 | . 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 6733 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnfc 2751 ↦ cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: cbvmpt2v 6735 el2mpt2csbcl 7250 fnmpt2ovd 7252 fmpt2co 7260 mpt2curryd 7395 fvmpt2curryd 7397 xpf1o 8122 cnfcomlem 8596 fseqenlem1 8847 relexpsucnnr 13765 gsumdixp 18609 evlslem4 19508 madugsum 20449 cnmpt2t 21476 cnmptk2 21489 fmucnd 22096 fsum2cn 22674 fmuldfeqlem1 39814 smflim 40985 |
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