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Mirrors > Home > ILE Home > Th. List > cbvmpov | GIF version |
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 4023, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.) |
Ref | Expression |
---|---|
cbvmpov.1 | ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸) |
cbvmpov.2 | ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷) |
Ref | Expression |
---|---|
cbvmpov | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2281 | . 2 ⊢ Ⅎ𝑧𝐶 | |
2 | nfcv 2281 | . 2 ⊢ Ⅎ𝑤𝐶 | |
3 | nfcv 2281 | . 2 ⊢ Ⅎ𝑥𝐷 | |
4 | nfcv 2281 | . 2 ⊢ Ⅎ𝑦𝐷 | |
5 | cbvmpov.1 | . . 3 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸) | |
6 | cbvmpov.2 | . . 3 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷) | |
7 | 5, 6 | sylan9eq 2192 | . 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) |
8 | 1, 2, 3, 4, 7 | cbvmpo 5850 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ cmpo 5776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-oprab 5778 df-mpo 5779 |
This theorem is referenced by: frec2uzrdg 10185 frecuzrdgsuc 10190 iseqvalcbv 10233 resqrexlemfp1 10784 resqrex 10801 sqne2sq 11858 ennnfonelemnn0 11938 txbas 12430 xmetxp 12679 |
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