Theorem dfima3 6092

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.)

Assertion

Ref	Expression
dfima3	⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem dfima3

Step	Hyp	Ref	Expression
1		dfima2 6091	. 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦}
2		df-br 5167	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	2	rexbii 3100	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥 ∈ 𝐵 ⟨𝑥, 𝑦⟩ ∈ 𝐴)
4		df-rex 3077	. . . 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 2812	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
7	1, 6	eqtri 2768	1 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∃wrex 3076  ⟨cop 4654   class class class wbr 5166   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  imadmrn  6099  imassrn  6100  imai  6103  funimaexgOLD  6665  cnvimadfsn  8213  rdglim2  8488  fineqvrep  35071  dfhe3  43737