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Mirrors > Home > ILE Home > Th. List > rneq | GIF version |
Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996.) |
Ref | Expression |
---|---|
rneq | ⊢ (A = B → ran A = ran B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 4452 | . . 3 ⊢ (A = B → ◡A = ◡B) | |
2 | 1 | dmeqd 4480 | . 2 ⊢ (A = B → dom ◡A = dom ◡B) |
3 | df-rn 4299 | . 2 ⊢ ran A = dom ◡A | |
4 | df-rn 4299 | . 2 ⊢ ran B = dom ◡B | |
5 | 2, 3, 4 | 3eqtr4g 2094 | 1 ⊢ (A = B → ran A = ran B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ◡ccnv 4287 dom cdm 4288 ran crn 4289 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-cnv 4296 df-dm 4298 df-rn 4299 |
This theorem is referenced by: rneqi 4505 rneqd 4506 xpima1 4710 feq1 4973 foeq1 5045 |
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