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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 2903 | . 2 ⊢ Ⅎ𝑧𝐵 | |
2 | nfcv 2903 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | cbvmpo.1 | . 2 ⊢ Ⅎ𝑧𝐶 | |
4 | cbvmpo.2 | . 2 ⊢ Ⅎ𝑤𝐶 | |
5 | cbvmpo.3 | . 2 ⊢ Ⅎ𝑥𝐷 | |
6 | cbvmpo.4 | . 2 ⊢ Ⅎ𝑦𝐷 | |
7 | eqidd 2736 | . 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 7526 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnfc 2888 ∈ cmpo 7433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: fvmpopr2d 7595 el2mpocsbcl 8109 fnmpoovd 8111 fmpoco 8119 mpocurryd 8293 fvmpocurryd 8295 xpf1o 9178 cnfcomlem 9737 fseqenlem1 10062 relexpsucnnr 15061 gsumdixp 20333 evlslem4 22118 madugsum 22665 cnmpt2t 23697 cnmptk2 23710 fmucnd 24317 fsum2cn 24909 aks6d1c7lem3 42164 fmpocos 42254 fmuldfeqlem1 45538 smflim 46733 |
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