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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 4783 | . . 3 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) | |
2 | 1 | dmeqd 4811 | . 2 ⊢ (𝐴 = 𝐵 → dom ◡𝐴 = dom ◡𝐵) |
3 | df-rn 4620 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
4 | df-rn 4620 | . 2 ⊢ ran 𝐵 = dom ◡𝐵 | |
5 | 2, 3, 4 | 3eqtr4g 2228 | 1 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ◡ccnv 4608 dom cdm 4609 ran crn 4610 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-cnv 4617 df-dm 4619 df-rn 4620 |
This theorem is referenced by: rneqi 4837 rneqd 4838 xpima1 5055 feq1 5328 foeq1 5414 ixpsnf1o 6711 |
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