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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 4838 | . 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 4612 |
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 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-cnv 4619 df-dm 4621 df-rn 4622 |
This theorem is referenced by: resima2 4925 imaeq1 4948 imaeq2 4949 resiima 4969 elxp4 5098 elxp5 5099 funimacnv 5274 funimaexg 5282 fnima 5316 fnrnfv 5543 2ndvalg 6122 fo2nd 6137 f2ndres 6139 en1 6777 xpassen 6808 xpdom2 6809 sbthlemi4 6937 djudom 7070 exmidfodomrlemim 7178 seqeq1 10404 seqeq2 10405 seqeq3 10406 seq3val 10414 seqvalcd 10415 ennnfonelemex 12369 ennnfonelemf1 12373 restval 12585 restid2 12588 tgrest 12963 txvalex 13048 txval 13049 mopnval 13236 |
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