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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 2228 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2228 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvmptv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvmpt 3933 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1289 ↦ cmpt 3899 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-un 3003 df-sn 3452 df-pr 3453 df-op 3455 df-opab 3900 df-mpt 3901 |
This theorem is referenced by: frecsuc 6172 xpmapen 6564 fodjuomni 6802 caucvgsrlembnd 7344 negiso 8414 infrenegsupex 9080 frec2uzsucd 9804 frecuzrdgdom 9821 frecuzrdgfun 9823 frecuzrdgsuct 9827 0tonninf 9841 1tonninf 9842 seq3f1oleml 9928 seq3f1o 9929 hashfz1 10187 climcvg1n 10735 isummo 10769 zisum 10770 fisum 10774 fsumadd 10796 phimullem 11475 nninfsellemqall 11862 nninfomni 11866 exmidsbthrlem 11867 exmidsbthr 11868 |
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