![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3228 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} ⊆ {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴} |
3 | dfima3 4974 | . 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} | |
4 | dfrn3 4817 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴} | |
5 | 2, 3, 4 | 3sstr4i 3197 | 1 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∃wex 1492 ∈ wcel 2148 {cab 2163 ⊆ wss 3130 ⟨cop 3596 ran crn 4628 “ cima 4630 |
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 4122 ax-pow 4175 ax-pr 4210 |
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 2740 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-br 4005 df-opab 4066 df-xp 4633 df-cnv 4635 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 |
This theorem is referenced by: imaexg 4983 0ima 4989 cnvimass 4992 fimacnv 5646 f1opw2 6077 smores2 6295 ecss 6576 f1imaen2g 6793 fopwdom 6836 ssenen 6851 phplem4dom 6862 isinfinf 6897 fiintim 6928 sbthlem2 6957 sbthlemi3 6958 sbthlemi5 6960 sbthlemi6 6961 ctssdccl 7110 ctinf 12431 ssnnctlemct 12447 mhmima 12875 cnptoprest2 13743 hmeontr 13816 hmeores 13818 tgqioo 14050 |
Copyright terms: Public domain | W3C validator |