Theorem elima3 6070

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 6069	. 2 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴))
3		df-br 5114	. . . 4 ⊢ (𝑥𝐵𝐴 ↔ ⟨𝑥, 𝐴⟩ ∈ 𝐵)
4	3	anbi2i 634	. . 3 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴) ↔ (𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
5	4	exbii 1875	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐶 ∧ 𝑥𝐵𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))
6	2, 5	bitri 278	1 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥(𝑥 ∈ 𝐶 ∧ ⟨𝑥, 𝐴⟩ ∈ 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1806   ∈ wcel 2149  Vcvv 3463  ⟨cop 4600   class class class wbr 5113   “ cima 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675

This theorem is referenced by:  cnvresima  6232  imaiun  7244  1stpreimas  32991  elima4  36166  imaiun1  44268  snhesn  44403