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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 4836 | . 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 1348 ran crn 4610 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-cnv 4617 df-dm 4619 df-rn 4620 |
This theorem is referenced by: resima2 4923 imaeq1 4946 imaeq2 4947 resiima 4967 elxp4 5096 elxp5 5097 funimacnv 5272 funimaexg 5280 fnima 5314 fnrnfv 5541 2ndvalg 6119 fo2nd 6134 f2ndres 6136 en1 6773 xpassen 6804 xpdom2 6805 sbthlemi4 6933 djudom 7066 exmidfodomrlemim 7165 seqeq1 10391 seqeq2 10392 seqeq3 10393 seq3val 10401 seqvalcd 10402 ennnfonelemex 12356 ennnfonelemf1 12360 restval 12572 restid2 12575 tgrest 12922 txvalex 13007 txval 13008 mopnval 13195 |
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