Theorem elima3 6056

Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 14-Aug-1994.)

Hypothesis

Ref	Expression
elima.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elima3	⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))

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

Proof of Theorem elima3

Step	Hyp	Ref	Expression
1		elima.1	. . 3 ⊢ 𝐴 ∈ V
2	1	elima2 6055	. 2 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴))
3		df-br 5101	. . . 4 ⊢ (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
4	3	anbi2i 632	. . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
5	4	exbii 1868	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
6	2, 5	bitri 277	1 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 399  ∃wex 1799   ∈ wcel 2142  Vcvv 3454  ⟨cop 4588   class class class wbr 5100   “ cima 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-cnv 5655  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660

This theorem is referenced by:  cnvresima  6217  imaiun  7229  1stpreimas  32905  elima4  36123  imaiun1  44224  snhesn  44359