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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 1605 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) → ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | |
2 | 1 | ss2abi 3209 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} ⊆ {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
3 | dfima3 4943 | . 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
4 | dfrn3 4787 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} | |
5 | 2, 3, 4 | 3sstr4i 3178 | 1 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∃wex 1479 ∈ wcel 2135 {cab 2150 ⊆ wss 3111 〈cop 3573 ran crn 4599 “ cima 4601 |
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-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 |
This theorem is referenced by: imaexg 4952 0ima 4958 cnvimass 4961 fimacnv 5608 f1opw2 6038 smores2 6253 ecss 6533 f1imaen2g 6750 fopwdom 6793 ssenen 6808 phplem4dom 6819 isinfinf 6854 fiintim 6885 sbthlem2 6914 sbthlemi3 6915 sbthlemi5 6917 sbthlemi6 6918 ctssdccl 7067 ctinf 12306 ssnnctlemct 12322 cnptoprest2 12787 hmeontr 12860 hmeores 12862 tgqioo 13094 |
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