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Mirrors > Home > MPE Home > Th. List > cbvmpo | 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 |
---|---|
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 2897 | . 2 ⊢ Ⅎ𝑧𝐵 | |
2 | nfcv 2897 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | cbvmpo.1 | . 2 ⊢ Ⅎ𝑧𝐶 | |
4 | cbvmpo.2 | . 2 ⊢ Ⅎ𝑤𝐶 | |
5 | cbvmpo.3 | . 2 ⊢ Ⅎ𝑥𝐷 | |
6 | cbvmpo.4 | . 2 ⊢ Ⅎ𝑦𝐷 | |
7 | eqidd 2737 | . 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 7282 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 Ⅎwnfc 2877 ∈ cmpo 7193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-opab 5102 df-oprab 7195 df-mpo 7196 |
This theorem is referenced by: cbvmpov 7284 fvmpopr2d 7348 el2mpocsbcl 7831 fnmpoovd 7833 fmpoco 7841 mpocurryd 7989 fvmpocurryd 7991 xpf1o 8786 cnfcomlem 9292 fseqenlem1 9603 relexpsucnnr 14553 gsumdixp 19581 evlslem4 20988 madugsum 21494 cnmpt2t 22524 cnmptk2 22537 fmucnd 23143 fsum2cn 23722 fmuldfeqlem1 42741 smflim 43927 |
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