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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 2660 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 2660 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opeldm 4702 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑥 ∈ dom 𝐴) |
4 | 3 | pm4.71i 386 | . . . . 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 1567 | . . 3 ⊢ (∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
8 | 7 | abbii 2230 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
9 | dfima3 4842 | . 2 ⊢ (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
10 | dfrn3 4688 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} | |
11 | 8, 9, 10 | 3eqtr4i 2145 | 1 ⊢ (𝐴 “ dom 𝐴) = ran 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1314 ∃wex 1451 ∈ wcel 1463 {cab 2101 〈cop 3496 dom cdm 4499 ran crn 4500 “ cima 4502 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-br 3896 df-opab 3950 df-xp 4505 df-cnv 4507 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 |
This theorem is referenced by: cnvimarndm 4861 foima 5308 f1imacnv 5340 fsn2 5548 resfunexg 5595 funiunfvdm 5618 fnexALT 5965 uniqs2 6443 mapsn 6538 phplem4 6702 phplem4on 6714 retopbas 12512 |
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