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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 1618 | . . 3 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) → ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) | |
2 | 1 | ss2abi 3227 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} ⊆ {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
3 | dfima3 4972 | . 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
4 | dfrn3 4815 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} | |
5 | 2, 3, 4 | 3sstr4i 3196 | 1 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∃wex 1492 ∈ wcel 2148 {cab 2163 ⊆ wss 3129 〈cop 3595 ran crn 4626 “ cima 4628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-br 4003 df-opab 4064 df-xp 4631 df-cnv 4633 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 |
This theorem is referenced by: imaexg 4981 0ima 4987 cnvimass 4990 fimacnv 5644 f1opw2 6074 smores2 6292 ecss 6573 f1imaen2g 6790 fopwdom 6833 ssenen 6848 phplem4dom 6859 isinfinf 6894 fiintim 6925 sbthlem2 6954 sbthlemi3 6955 sbthlemi5 6957 sbthlemi6 6958 ctssdccl 7107 ctinf 12423 ssnnctlemct 12439 mhmima 12807 cnptoprest2 13611 hmeontr 13684 hmeores 13686 tgqioo 13918 |
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