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| Mirrors > Home > MPE Home > Th. List > imaeq2 | Structured version Visualization version GIF version | ||
| Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.) |
| Ref | Expression |
|---|---|
| imaeq2 | ⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reseq2 5992 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | |
| 2 | 1 | rneqd 5949 | . 2 ⊢ (𝐴 = 𝐵 → ran (𝐶 ↾ 𝐴) = ran (𝐶 ↾ 𝐵)) |
| 3 | df-ima 5698 | . 2 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴) | |
| 4 | df-ima 5698 | . 2 ⊢ (𝐶 “ 𝐵) = ran (𝐶 ↾ 𝐵) | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝐴 = 𝐵 → (𝐶 “ 𝐴) = (𝐶 “ 𝐵)) |
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