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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 3482 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 3482 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opeldm 5921 | . . . . . 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 1845 | . . 3 ⊢ (∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
8 | 7 | abbii 2807 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
9 | dfima3 6083 | . 2 ⊢ (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
10 | dfrn3 5903 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} | |
11 | 8, 9, 10 | 3eqtr4i 2773 | 1 ⊢ (𝐴 “ dom 𝐴) = ran 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 〈cop 4637 dom cdm 5689 ran crn 5690 “ cima 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: cnvimarndm 6103 foima 6826 fimadmfo 6830 f1imacnv 6865 fsn2 7156 resfunexg 7235 elunirnALT 7272 fnexALT 7974 uniqs2 8818 mapsnd 8925 phplem2 9243 php3 9247 phplem4OLD 9255 php3OLD 9259 pwfilem 9354 jech9.3 9852 fin4en1 10347 retopbas 24797 plyeq0 26265 bday0s 27888 rnelshi 32088 s2rnOLD 32913 s3rnOLD 32915 rndrhmcl 33280 qusrn 33417 rhmimaidl 33440 ply1degltdimlem 33650 poimirlem3 37610 poimirlem30 37637 |
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