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Mirrors > Home > ILE Home > Th. List > cbvmptv | GIF version |
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by Mario Carneiro, 19-Feb-2013.) |
Ref | Expression |
---|---|
cbvmptv.1 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvmptv | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2178 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2178 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvmptv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvmpt 3851 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ↦ cmpt 3818 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-opab 3819 df-mpt 3820 |
This theorem is referenced by: frecsuc 5991 caucvgsrlembnd 6885 frec2uzzd 9186 frec2uzsucd 9187 climcvg1n 9869 |
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