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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 5968 | . 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} | |
2 | df-br 5079 | . . . . 5 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | 2 | rexbii 3179 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥 ∈ 𝐵 〈𝑥, 𝑦〉 ∈ 𝐴) |
4 | df-rex 3071 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 〈𝑥, 𝑦〉 ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) | |
5 | 3, 4 | bitri 274 | . . 3 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)) |
6 | 5 | abbii 2809 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} |
7 | 1, 6 | eqtri 2767 | 1 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1541 ∃wex 1785 ∈ wcel 2109 {cab 2716 ∃wrex 3066 〈cop 4572 class class class wbr 5078 “ cima 5591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-sb 2071 df-clab 2717 df-cleq 2731 df-clel 2817 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-dif 3894 df-un 3896 df-in 3898 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-br 5079 df-opab 5141 df-xp 5594 df-cnv 5596 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 |
This theorem is referenced by: imadmrn 5976 imassrn 5977 imai 5979 funimaexg 6516 cnvimadfsn 7972 rdglim2 8247 fineqvrep 33043 dfhe3 41336 |
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