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Mirrors > Home > MPE Home > Th. List > dfima2 | Structured version Visualization version 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.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
Ref | Expression |
---|---|
dfima2 | ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 5532 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | dfrn2 5723 | . 2 ⊢ ran (𝐴 ↾ 𝐵) = {𝑦 ∣ ∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦} | |
3 | brres 5825 | . . . . . 6 ⊢ (𝑦 ∈ V → (𝑥(𝐴 ↾ 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦))) | |
4 | 3 | elv 3446 | . . . . 5 ⊢ (𝑥(𝐴 ↾ 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) |
5 | 4 | exbii 1849 | . . . 4 ⊢ (∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) |
6 | df-rex 3112 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) | |
7 | 5, 6 | bitr4i 281 | . . 3 ⊢ (∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦 ↔ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦) |
8 | 7 | abbii 2863 | . 2 ⊢ {𝑦 ∣ ∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} |
9 | 1, 2, 8 | 3eqtri 2825 | 1 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 {cab 2776 ∃wrex 3107 Vcvv 3441 class class class wbr 5030 ran crn 5520 ↾ cres 5521 “ cima 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: dfima3 5899 elimag 5900 imasng 5918 dfimafn 6703 isoini 7070 dffin1-5 9799 dfimafnf 30395 ofpreima 30428 dfaimafn 43721 |
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