![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfima3 | Structured version Visualization version GIF version |
Description: Alternate definition of image. Compare definition (d) of [Enderton] p. 44. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dfima3 | ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfima2 6052 | . 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} | |
2 | df-br 5140 | . . . . 5 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
3 | 2 | rexbii 3086 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥 ∈ 𝐵 ⟨𝑥, 𝑦⟩ ∈ 𝐴) |
4 | df-rex 3063 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)) | |
5 | 3, 4 | bitri 275 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)) |
6 | 5 | abbii 2794 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} |
7 | 1, 6 | eqtri 2752 | 1 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2701 ∃wrex 3062 ⟨cop 4627 class class class wbr 5139 “ cima 5670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 |
This theorem is referenced by: imadmrn 6060 imassrn 6061 imai 6064 funimaexgOLD 6626 cnvimadfsn 8152 rdglim2 8428 fineqvrep 34613 dfhe3 43075 |
Copyright terms: Public domain | W3C validator |