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Mirrors > Home > ILE Home > Th. List > imassrn | 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 1509 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) → ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | |
2 | 1 | ss2abi 3012 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} ⊆ {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
3 | dfima3 4671 | . 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
4 | dfrn3 4524 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} | |
5 | 2, 3, 4 | 3sstr4i 2984 | 1 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ∃wex 1381 ∈ wcel 1393 {cab 2026 ⊆ wss 2917 〈cop 3378 ran crn 4346 “ cima 4348 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 |
This theorem is referenced by: imaexg 4680 0ima 4685 cnvimass 4688 fimacnv 5296 f1opw2 5706 smores2 5909 ecss 6147 f1imaen2g 6273 fopwdom 6310 phplem4dom 6324 |
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