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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 4774 | . 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 1332 ran crn 4548 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-cnv 4555 df-dm 4557 df-rn 4558 |
This theorem is referenced by: rnmpt 4795 resima 4860 resima2 4861 ima0 4906 rnuni 4958 imaundi 4959 imaundir 4960 inimass 4963 dminxp 4991 imainrect 4992 xpima1 4993 xpima2m 4994 rnresv 5006 imacnvcnv 5011 rnpropg 5026 imadmres 5039 mptpreima 5040 dmco 5055 resdif 5397 fpr 5610 fprg 5611 fliftfuns 5707 rnoprab 5862 rnmpo 5889 qliftfuns 6521 xpassen 6732 sbthlemi6 6858 ennnfonelemrn 11968 cnconst2 12441 |
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