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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 2733 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 2733 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opeldm 4814 | . . . . . 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 1598 | . . 3 ⊢ (∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴) |
8 | 7 | abbii 2286 | . 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} |
9 | dfima3 4956 | . 2 ⊢ (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
10 | dfrn3 4800 | . 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥〈𝑥, 𝑦〉 ∈ 𝐴} | |
11 | 8, 9, 10 | 3eqtr4i 2201 | 1 ⊢ (𝐴 “ dom 𝐴) = ran 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ∃wex 1485 ∈ wcel 2141 {cab 2156 〈cop 3586 dom cdm 4611 ran crn 4612 “ cima 4614 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 |
This theorem is referenced by: cnvimarndm 4975 foima 5425 f1imacnv 5459 fsn2 5670 resfunexg 5717 funiunfvdm 5742 fnexALT 6090 uniqs2 6573 mapsn 6668 phplem4 6833 phplem4on 6845 retopbas 13317 |
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