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Mirrors > Home > ILE Home > Th. List > dfima2 | GIF version |
Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 19-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dfima2 | ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4376 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | dfrn2 4541 | . 2 ⊢ ran (𝐴 ↾ 𝐵) = {𝑦 ∣ ∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦} | |
3 | vex 2604 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
4 | 3 | brres 4636 | . . . . . 6 ⊢ (𝑥(𝐴 ↾ 𝐵)𝑦 ↔ (𝑥𝐴𝑦 ∧ 𝑥 ∈ 𝐵)) |
5 | ancom 262 | . . . . . 6 ⊢ ((𝑥𝐴𝑦 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) | |
6 | 4, 5 | bitri 182 | . . . . 5 ⊢ (𝑥(𝐴 ↾ 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) |
7 | 6 | exbii 1536 | . . . 4 ⊢ (∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) |
8 | df-rex 2354 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) | |
9 | 7, 8 | bitr4i 185 | . . 3 ⊢ (∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦 ↔ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦) |
10 | 9 | abbii 2194 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} |
11 | 1, 2, 10 | 3eqtri 2105 | 1 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1284 ∃wex 1421 ∈ wcel 1433 {cab 2067 ∃wrex 2349 class class class wbr 3785 ran crn 4364 ↾ cres 4365 “ cima 4366 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-cnv 4371 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 |
This theorem is referenced by: dfima3 4691 elimag 4692 imasng 4710 imadiflem 4998 imadif 4999 imainlem 5000 imain 5001 funimaexglem 5002 dfimafn 5243 isoini 5477 |
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