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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 5325 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | dfrn2 5514 | . 2 ⊢ ran (𝐴 ↾ 𝐵) = {𝑦 ∣ ∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦} | |
3 | brres 5607 | . . . . . 6 ⊢ (𝑦 ∈ V → (𝑥(𝐴 ↾ 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦))) | |
4 | 3 | elv 3389 | . . . . 5 ⊢ (𝑥(𝐴 ↾ 𝐵)𝑦 ↔ (𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) |
5 | 4 | exbii 1944 | . . . 4 ⊢ (∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) |
6 | df-rex 3095 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑥𝐴𝑦)) | |
7 | 5, 6 | bitr4i 270 | . . 3 ⊢ (∃𝑥 𝑥(𝐴 ↾ 𝐵)𝑦 ↔ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦) |
8 | 7 | abbii 2916 | . 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 198 ∧ wa 385 = wceq 1653 ∃wex 1875 ∈ wcel 2157 {cab 2785 ∃wrex 3090 Vcvv 3385 class class class wbr 4843 ran crn 5313 ↾ cres 5314 “ cima 5315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-sn 4369 df-pr 4371 df-op 4375 df-br 4844 df-opab 4906 df-xp 5318 df-cnv 5320 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 |
This theorem is referenced by: dfima3 5686 elimag 5687 imasng 5704 dfimafn 6470 isoini 6816 dffin1-5 9498 dfimafnf 29955 ofpreima 29984 dfaimafn 42019 |
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