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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 2319 | . 2 ⊢ Ⅎ𝑦𝐵 | |
2 | nfcv 2319 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | cbvmptv.1 | . 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | cbvmpt 4095 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ↦ cmpt 4061 |
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-ext 2159 |
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 2739 df-un 3133 df-sn 3597 df-pr 3598 df-op 3600 df-opab 4062 df-mpt 4063 |
This theorem is referenced by: fnmptfvd 5615 frecsuc 6401 xpmapen 6843 omp1eom 7087 fodjuomni 7140 fodjumkv 7151 nninfwlporlemd 7163 nninfwlpor 7165 nninfwlpoim 7169 caucvgsrlembnd 7778 negiso 8888 infrenegsupex 9570 frec2uzsucd 10374 frecuzrdgdom 10391 frecuzrdgfun 10393 frecuzrdgsuct 10397 0tonninf 10412 1tonninf 10413 seq3f1oleml 10476 seq3f1o 10477 hashfz1 10734 xrnegiso 11241 infxrnegsupex 11242 climcvg1n 11329 summodc 11362 zsumdc 11363 fsum3 11366 fsumadd 11385 prodmodc 11557 zproddc 11558 fprodseq 11562 phimullem 12195 eulerthlemh 12201 eulerthlemth 12202 ennnfonelemnn0 12393 ennnfonelemr 12394 ctinfom 12399 grplactcnv 12848 cdivcncfap 13720 expcncf 13725 2sqlem1 14083 bj-charfunbi 14185 subctctexmid 14373 nninfsellemqall 14387 nninfomni 14391 nninffeq 14392 exmidsbthrlem 14393 exmidsbthr 14394 isomninn 14402 iswomninn 14421 ismkvnn 14424 |
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