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Mirrors > Home > ILE Home > Th. List > dmeqd | GIF version |
Description: Equality deduction for domain. (Contributed by NM, 4-Mar-2004.) |
Ref | Expression |
---|---|
dmeqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
dmeqd | ⊢ (𝜑 → dom 𝐴 = dom 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | dmeq 4739 | . 2 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → dom 𝐴 = dom 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 dom cdm 4539 |
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-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-dm 4549 |
This theorem is referenced by: rneq 4766 dmsnsnsng 5016 elxp4 5026 fndmin 5527 1stvalg 6040 fo1st 6055 f1stres 6057 errn 6451 xpassen 6724 xpdom2 6725 frecuzrdgtclt 10194 shftdm 10594 ennnfonelemg 11916 ennnfonelem1 11920 ennnfonelemhdmp1 11922 ennnfonelemkh 11925 ennnfonelemhf1o 11926 ennnfonelemex 11927 ennnfonelemhom 11928 isstruct2im 11969 isstruct2r 11970 setsvalg 11989 cnprcl2k 12375 psmetdmdm 12493 xmetdmdm 12525 blfvalps 12554 limccl 12797 ellimc3apf 12798 dvfvalap 12819 dvcj 12842 dvexp 12844 dvmptclx 12849 dvmptaddx 12850 dvmptmulx 12851 |
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