Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2977 | . 2 ⊢ Ⅎ𝑧𝐵 | |
2 | nfcv 2977 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | cbvmpo.1 | . 2 ⊢ Ⅎ𝑧𝐶 | |
4 | cbvmpo.2 | . 2 ⊢ Ⅎ𝑤𝐶 | |
5 | cbvmpo.3 | . 2 ⊢ Ⅎ𝑥𝐷 | |
6 | cbvmpo.4 | . 2 ⊢ Ⅎ𝑦𝐷 | |
7 | eqidd 2822 | . 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 7247 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnfc 2961 ∈ cmpo 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: cbvmpov 7249 el2mpocsbcl 7780 fnmpoovd 7782 fmpoco 7790 mpocurryd 7935 fvmpocurryd 7937 xpf1o 8679 cnfcomlem 9162 fseqenlem1 9450 relexpsucnnr 14384 gsumdixp 19359 evlslem4 20288 madugsum 21252 cnmpt2t 22281 cnmptk2 22294 fmucnd 22901 fsum2cn 23479 fmuldfeqlem1 41912 smflim 43102 |
Copyright terms: Public domain | W3C validator |