Theorem elimag 6057

Description: Membership in an image. Theorem 34 of [Suppes] p. 65. (Contributed by NM, 20-Jan-2007.)

Assertion

Ref	Expression
elimag	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))

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

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elimag

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5109	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝐴))
2	1	rexbidv 3189	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑥𝐵𝑦 ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))
3		dfima2 6055	. 2 ⊢ (𝐵 “ 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑥𝐵𝑦}
4	2, 3	elab2g 3642	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1563   ∈ wcel 2145  ∃wrex 3089   class class class wbr 5105   “ cima 5655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665

This theorem is referenced by:  elima  6058  fvelima2  6923  fvelima  6936  fvelimad  6938  opelco3  36138  afvelima  47759  inisegn0a  49465