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Mirrors > Home > ILE Home > Th. List > imadmrn | 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 2724 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 2724 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opeldm 4801 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
4 | 3 | pm4.71i 389 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ (〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐴)) |
5 | ancom 264 | . . . . 5 ⊢ ((〈𝑥, 𝑦〉 ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
6 | 4, 5 | bitr2i 184 | . . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) |
7 | 6 | exbii 1592 | . . 3 ⊢ (∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
8 | 7 | abbii 2280 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
9 | dfima3 4943 | . 2 ⊢ (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
10 | dfrn3 4787 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} | |
11 | 8, 9, 10 | 3eqtr4i 2195 | 1 ⊢ (𝐴 “ dom 𝐴) = ran 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∃wex 1479 ∈ wcel 2135 {cab 2150 〈cop 3573 dom cdm 4598 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: cnvimarndm 4962 foima 5409 f1imacnv 5443 fsn2 5653 resfunexg 5700 funiunfvdm 5725 fnexALT 6073 uniqs2 6552 mapsn 6647 phplem4 6812 phplem4on 6824 retopbas 13070 |
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