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Mirrors > Home > ILE Home > Th. List > dmeqi | GIF version |
Description: Equality inference for domain. (Contributed by NM, 4-Mar-2004.) |
Ref | Expression |
---|---|
dmeqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
dmeqi | ⊢ dom 𝐴 = dom 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | dmeq 4734 | . 2 ⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ dom 𝐴 = dom 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 dom cdm 4534 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-dm 4544 |
This theorem is referenced by: dmxpm 4754 dmxpid 4755 dmxpin 4756 rncoss 4804 rncoeq 4807 rnun 4942 rnin 4943 rnxpm 4963 rnxpss 4965 imainrect 4979 dmpropg 5006 dmtpop 5009 rnsnopg 5012 fntpg 5174 fnreseql 5523 dmoprab 5845 reldmmpo 5875 elmpocl 5961 tfrlem8 6208 tfr2a 6211 tfrlemi14d 6223 tfr1onlemres 6239 tfri1dALT 6241 tfrcllemres 6252 xpassen 6717 sbthlemi5 6842 casedm 6964 djudm 6983 ctssdccl 6989 dmaddpi 7126 dmmulpi 7127 dmaddpq 7180 dmmulpq 7181 axaddf 7669 axmulf 7670 ennnfonelemom 11910 ennnfonelemdm 11922 |
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