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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 4099, 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 2319 | . 2 ⊢ Ⅎ𝑧𝐶 | |
2 | nfcv 2319 | . 2 ⊢ Ⅎ𝑤𝐶 | |
3 | nfcv 2319 | . 2 ⊢ Ⅎ𝑥𝐷 | |
4 | nfcv 2319 | . 2 ⊢ Ⅎ𝑦𝐷 | |
5 | cbvmpov.1 | . . 3 ⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸) | |
6 | cbvmpov.2 | . . 3 ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷) | |
7 | 5, 6 | sylan9eq 2230 | . 2 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) |
8 | 1, 2, 3, 4, 7 | cbvmpo 5954 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ cmpo 5877 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4122 ax-pow 4175 ax-pr 4210 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-opab 4066 df-oprab 5879 df-mpo 5880 |
This theorem is referenced by: frec2uzrdg 10409 frecuzrdgsuc 10414 iseqvalcbv 10457 resqrexlemfp1 11018 resqrex 11035 sqne2sq 12177 ennnfonelemnn0 12423 nninfdc 12454 txbas 13761 xmetxp 14010 |
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