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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 2281 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2281 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvmptv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvmpt 4023 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ↦ cmpt 3989 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
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-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-mpt 3991 |
This theorem is referenced by: frecsuc 6304 xpmapen 6744 omp1eom 6980 fodjuomni 7021 fodjumkv 7034 caucvgsrlembnd 7609 negiso 8713 infrenegsupex 9389 frec2uzsucd 10174 frecuzrdgdom 10191 frecuzrdgfun 10193 frecuzrdgsuct 10197 0tonninf 10212 1tonninf 10213 seq3f1oleml 10276 seq3f1o 10277 hashfz1 10529 xrnegiso 11031 infxrnegsupex 11032 climcvg1n 11119 summodc 11152 zsumdc 11153 fsum3 11156 fsumadd 11175 prodmodc 11347 phimullem 11901 ennnfonelemnn0 11935 ennnfonelemr 11936 ctinfom 11941 cdivcncfap 12756 expcncf 12761 subctctexmid 13196 nninfsellemqall 13211 nninfomni 13215 nninffeq 13216 exmidsbthrlem 13217 exmidsbthr 13218 isomninn 13226 |
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