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Mirrors > Home > ILE Home > Th. List > rneqi | GIF version |
Description: Equality inference for range. (Contributed by NM, 4-Mar-2004.) |
Ref | Expression |
---|---|
rneqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
rneqi | ⊢ ran 𝐴 = ran 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rneqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | rneq 4736 | . 2 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ran 𝐴 = ran 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ran crn 4510 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-cnv 4517 df-dm 4519 df-rn 4520 |
This theorem is referenced by: rnmpt 4757 resima 4822 resima2 4823 ima0 4868 rnuni 4920 imaundi 4921 imaundir 4922 inimass 4925 dminxp 4953 imainrect 4954 xpima1 4955 xpima2m 4956 rnresv 4968 imacnvcnv 4973 rnpropg 4988 imadmres 5001 mptpreima 5002 dmco 5017 resdif 5357 fpr 5570 fprg 5571 fliftfuns 5667 rnoprab 5822 rnmpo 5849 qliftfuns 6481 xpassen 6692 sbthlemi6 6818 ennnfonelemrn 11859 cnconst2 12329 |
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