Theorem elima 6079

Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 19-Apr-2004.)

Hypothesis

Ref	Expression
elima.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elima	⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴)

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

Proof of Theorem elima

Step	Hyp	Ref	Expression
1		elima.1	. 2 ⊢ 𝐴 ∈ V
2		elimag 6078	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   class class class wbr 5108   “ cima 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671

This theorem is used by:  elima2  6080  rninxp  6189  imaco  6271  imaindm  6328  isarep1  6658  eliman0  6953  funimass4  6980  isomin  7376  dfsup2  9473  dfac10b  10257  hausmapdom  23874  pi1blem  25415  cutsun12  28233  madeval2  28276  cgrabasimass  29435  adjbd1o  32744  brimage  36903  dfrecs2  36929  dfrdg4  36930  dfint3  36931  imagesset  36932  elimaint  44910  elintima  44914