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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 2976 | . 2 ⊢ Ⅎ𝑧𝐵 | |
2 | nfcv 2976 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | cbvmpo.1 | . 2 ⊢ Ⅎ𝑧𝐶 | |
4 | cbvmpo.2 | . 2 ⊢ Ⅎ𝑤𝐶 | |
5 | cbvmpo.3 | . 2 ⊢ Ⅎ𝑥𝐷 | |
6 | cbvmpo.4 | . 2 ⊢ Ⅎ𝑦𝐷 | |
7 | eqidd 2821 | . 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 7240 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 Ⅎwnfc 2960 ∈ cmpo 7151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5122 df-oprab 7153 df-mpo 7154 |
This theorem is referenced by: cbvmpov 7242 el2mpocsbcl 7773 fnmpoovd 7775 fmpoco 7783 mpocurryd 7928 fvmpocurryd 7930 xpf1o 8672 cnfcomlem 9155 fseqenlem1 9443 relexpsucnnr 14377 gsumdixp 19352 evlslem4 20281 madugsum 21245 cnmpt2t 22274 cnmptk2 22287 fmucnd 22894 fsum2cn 23472 fmuldfeqlem1 41938 smflim 43128 |
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