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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 4671 | . . 3 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) | |
2 | 1 | dmeqd 4699 | . 2 ⊢ (𝐴 = 𝐵 → dom ◡𝐴 = dom ◡𝐵) |
3 | df-rn 4508 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
4 | df-rn 4508 | . 2 ⊢ ran 𝐵 = dom ◡𝐵 | |
5 | 2, 3, 4 | 3eqtr4g 2170 | 1 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1312 ◡ccnv 4496 dom cdm 4497 ran crn 4498 |
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 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-sn 3497 df-pr 3498 df-op 3500 df-br 3894 df-opab 3948 df-cnv 4505 df-dm 4507 df-rn 4508 |
This theorem is referenced by: rneqi 4725 rneqd 4726 xpima1 4941 feq1 5211 foeq1 5297 ixpsnf1o 6582 |
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