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| Mirrors > Home > MPE Home > Th. List > imassrn | Structured version Visualization version GIF version | ||
| Description: The image of a class is a subset of its range. Theorem 3.16(xi) of [Monk1] p. 39. (Contributed by NM, 31-Mar-1995.) |
| Ref | Expression |
|---|---|
| imassrn | ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | exsimpr 1869 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) → ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | |
| 2 | 1 | ss2abi 4067 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} ⊆ {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
| 3 | dfima3 6081 | . 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
| 4 | dfrn3 5900 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} | |
| 5 | 2, 3, 4 | 3sstr4i 4035 | 1 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 |
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