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| Mirrors > Home > MPE Home > Th. List > imadmrn | Structured version Visualization version GIF version | ||
| Description: The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.) |
| Ref | Expression |
|---|---|
| imadmrn | ⊢ (𝐴 “ dom 𝐴) = ran 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | vex 3484 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
| 2 | vex 3484 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
| 3 | 1, 2 | opeldm 5918 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
| 4 | 3 | pm4.71i 559 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐴)) |
| 5 | ancom 460 | . . . . 5 ⊢ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
| 6 | 4, 5 | bitr2i 276 | . . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
| 7 | 6 | exbii 1848 | . . 3 ⊢ (∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
| 8 | 7 | abbii 2809 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
| 9 | dfima3 6081 | . 2 ⊢ (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
| 10 | dfrn3 5900 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} | |
| 11 | 8, 9, 10 | 3eqtr4i 2775 | 1 ⊢ (𝐴 “ dom 𝐴) = ran 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 〈cop 4632 dom cdm 5685 ran crn 5686 “ cima 5688 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: cnvimarndm 6101 f1imadifssran 6652 foima 6825 fimadmfo 6829 f1imacnv 6864 fsn2 7156 resfunexg 7235 elunirnALT 7272 fnexALT 7975 uniqs2 8819 mapsnd 8926 phplem2 9245 php3 9249 phplem4OLD 9257 php3OLD 9261 pwfilem 9356 jech9.3 9854 fin4en1 10349 retopbas 24781 plyeq0 26250 bday0s 27873 rnelshi 32078 s2rnOLD 32928 s3rnOLD 32930 rndrhmcl 33299 qusrn 33437 rhmimaidl 33460 ply1degltdimlem 33673 poimirlem3 37630 poimirlem30 37657 cycl3grtri 47914 |
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