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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 4876 | . 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 1372 dom cdm 4673 |
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-v 2773 df-un 3169 df-in 3171 df-ss 3178 df-sn 3638 df-pr 3639 df-op 3641 df-br 4044 df-dm 4683 |
| This theorem is referenced by: dmxpm 4896 dmxpid 4897 dmxpin 4898 rncoss 4946 rncoeq 4949 rnun 5088 rnin 5089 rnxpm 5109 rnxpss 5111 imainrect 5125 dmpropg 5152 dmtpop 5155 rnsnopg 5158 fntpg 5324 fnreseql 5684 dmoprab 6016 reldmmpo 6047 elmpocl 6131 tfrlem8 6394 tfr2a 6397 tfrlemi14d 6409 tfr1onlemres 6425 tfri1dALT 6427 tfrcllemres 6438 xpassen 6907 sbthlemi5 7045 casedm 7170 djudm 7189 ctssdccl 7195 dmaddpi 7420 dmmulpi 7421 dmaddpq 7474 dmmulpq 7475 axaddf 7963 axmulf 7964 ennnfonelemom 12698 ennnfonelemdm 12710 |
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