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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 4301 | . 2 ⊢ (A “ B) = ran (A ↾ B) | |
2 | dfrn2 4466 | . 2 ⊢ ran (A ↾ B) = {y ∣ ∃x x(A ↾ B)y} | |
3 | vex 2554 | . . . . . . 7 ⊢ y ∈ V | |
4 | 3 | brres 4561 | . . . . . 6 ⊢ (x(A ↾ B)y ↔ (xAy ∧ x ∈ B)) |
5 | ancom 253 | . . . . . 6 ⊢ ((xAy ∧ x ∈ B) ↔ (x ∈ B ∧ xAy)) | |
6 | 4, 5 | bitri 173 | . . . . 5 ⊢ (x(A ↾ B)y ↔ (x ∈ B ∧ xAy)) |
7 | 6 | exbii 1493 | . . . 4 ⊢ (∃x x(A ↾ B)y ↔ ∃x(x ∈ B ∧ xAy)) |
8 | df-rex 2306 | . . . 4 ⊢ (∃x ∈ B xAy ↔ ∃x(x ∈ B ∧ xAy)) | |
9 | 7, 8 | bitr4i 176 | . . 3 ⊢ (∃x x(A ↾ B)y ↔ ∃x ∈ B xAy) |
10 | 9 | abbii 2150 | . 2 ⊢ {y ∣ ∃x x(A ↾ B)y} = {y ∣ ∃x ∈ B xAy} |
11 | 1, 2, 10 | 3eqtri 2061 | 1 ⊢ (A “ B) = {y ∣ ∃x ∈ B xAy} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ∃wex 1378 ∈ wcel 1390 {cab 2023 ∃wrex 2301 class class class wbr 3755 ran crn 4289 ↾ cres 4290 “ cima 4291 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-cnv 4296 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 |
This theorem is referenced by: dfima3 4614 elimag 4615 imasng 4633 imadiflem 4921 imadif 4922 imainlem 4923 imain 4924 funimaexglem 4925 dfimafn 5165 isoini 5400 |
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