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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 2306 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2306 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvmptv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvmpt 4071 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ↦ cmpt 4037 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-opab 4038 df-mpt 4039 |
This theorem is referenced by: frecsuc 6366 xpmapen 6807 omp1eom 7051 fodjuomni 7104 fodjumkv 7115 caucvgsrlembnd 7733 negiso 8841 infrenegsupex 9523 frec2uzsucd 10326 frecuzrdgdom 10343 frecuzrdgfun 10345 frecuzrdgsuct 10349 0tonninf 10364 1tonninf 10365 seq3f1oleml 10428 seq3f1o 10429 hashfz1 10685 xrnegiso 11189 infxrnegsupex 11190 climcvg1n 11277 summodc 11310 zsumdc 11311 fsum3 11314 fsumadd 11333 prodmodc 11505 zproddc 11506 fprodseq 11510 phimullem 12136 eulerthlemh 12142 eulerthlemth 12143 ennnfonelemnn0 12298 ennnfonelemr 12299 ctinfom 12304 cdivcncfap 13134 expcncf 13139 bj-charfunbi 13534 subctctexmid 13722 nninfsellemqall 13736 nninfomni 13740 nninffeq 13741 exmidsbthrlem 13742 exmidsbthr 13743 isomninn 13751 iswomninn 13770 ismkvnn 13773 |
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