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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 4845 | . 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 1364 dom cdm 4644 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-dm 4654 |
This theorem is referenced by: rneq 4872 dmsnsnsng 5124 elxp4 5134 fndmin 5644 1stvalg 6168 fo1st 6183 f1stres 6185 errn 6582 xpassen 6857 xpdom2 6858 frecuzrdgtclt 10454 shftdm 10866 ennnfonelemg 12457 ennnfonelem1 12461 ennnfonelemhdmp1 12463 ennnfonelemkh 12466 ennnfonelemhf1o 12467 ennnfonelemex 12468 ennnfonelemhom 12469 isstruct2im 12525 isstruct2r 12526 setsvalg 12545 igsumvalx 12868 cnprcl2k 14183 psmetdmdm 14301 xmetdmdm 14333 blfvalps 14362 limccl 14605 ellimc3apf 14606 dvfvalap 14627 dvcj 14650 dvexp 14652 dvmptclx 14657 dvmptaddx 14658 dvmptmulx 14659 |
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