Theorem dfima3 5972

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 5971	. 2 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦}
2		df-br 5075	. . . . 5 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
3	2	rexbii 3181	. . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑥𝐴𝑦 ↔ ∃𝑥 ∈ 𝐵 ⟨𝑥, 𝑦⟩ ∈ 𝐴)
4		df-rex 3070	. . . 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 2808	. 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
7	1, 6	eqtri 2766	1 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐵 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}

Syntax hints:   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715  ∃wrex 3065  ⟨cop 4567   class class class wbr 5074   “ cima 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602

This theorem is referenced by:  imadmrn  5979  imassrn  5980  imai  5982  funimaexg  6520  cnvimadfsn  7988  rdglim2  8263  fineqvrep  33064  dfhe3  41383