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Mirrors > Home > ILE Home > Th. List > rneqd | GIF version |
Description: Equality deduction for range. (Contributed by NM, 4-Mar-2004.) |
Ref | Expression |
---|---|
rneqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
rneqd | ⊢ (𝜑 → ran 𝐴 = ran 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rneqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | rneq 4761 | . 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → ran 𝐴 = ran 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ran crn 4535 |
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-opab 3985 df-cnv 4542 df-dm 4544 df-rn 4545 |
This theorem is referenced by: resima2 4848 imaeq1 4871 imaeq2 4872 resiima 4892 elxp4 5021 elxp5 5022 funimacnv 5194 funimaexg 5202 fnima 5236 fnrnfv 5461 2ndvalg 6034 fo2nd 6049 f2ndres 6051 en1 6686 xpassen 6717 xpdom2 6718 sbthlemi4 6841 djudom 6971 exmidfodomrlemim 7050 seqeq1 10214 seqeq2 10215 seqeq3 10216 seq3val 10224 seqvalcd 10225 ennnfonelemex 11916 ennnfonelemf1 11920 restval 12115 restid2 12118 tgrest 12327 txvalex 12412 txval 12413 mopnval 12600 |
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