![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 4675 | . 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 1290 ran crn 4453 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-opab 3906 df-cnv 4460 df-dm 4462 df-rn 4463 |
This theorem is referenced by: resima2 4759 imaeq1 4782 imaeq2 4783 resiima 4803 elxp4 4931 elxp5 4932 funimacnv 5103 funimaexg 5111 fnima 5145 fnrnfv 5364 2ndvalg 5928 fo2nd 5943 f2ndres 5945 en1 6570 xpassen 6600 xpdom2 6601 sbthlemi4 6723 djudom 6837 exmidfodomrlemim 6888 iseqeq1 9919 iseqeq2 9920 iseqeq3 9921 iseqval 9932 iseqvalt 9934 seq3val 9935 restval 11719 restid2 11722 |
Copyright terms: Public domain | W3C validator |