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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 1549 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) → ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | |
2 | 1 | ss2abi 3066 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} ⊆ {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
3 | dfima3 4691 | . 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
4 | dfrn3 4542 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} | |
5 | 2, 3, 4 | 3sstr4i 3038 | 1 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ∃wex 1421 ∈ wcel 1433 {cab 2067 ⊆ wss 2973 〈cop 3401 ran crn 4364 “ cima 4366 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 |
This theorem is referenced by: imaexg 4700 0ima 4705 cnvimass 4708 fimacnv 5317 f1opw2 5726 smores2 5932 ecss 6170 f1imaen2g 6296 fopwdom 6333 phplem4dom 6348 |
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