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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 4772 | . . 3 ⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) | |
2 | 1 | dmeqd 4800 | . 2 ⊢ (𝐴 = 𝐵 → dom ◡𝐴 = dom ◡𝐵) |
3 | df-rn 4609 | . 2 ⊢ ran 𝐴 = dom ◡𝐴 | |
4 | df-rn 4609 | . 2 ⊢ ran 𝐵 = dom ◡𝐵 | |
5 | 2, 3, 4 | 3eqtr4g 2222 | 1 ⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ◡ccnv 4597 dom cdm 4598 ran crn 4599 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: rneqi 4826 rneqd 4827 xpima1 5044 feq1 5314 foeq1 5400 ixpsnf1o 6693 |
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