Theorem elimag 5973

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 5078	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝐴))
2	1	rexbidv 3226	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑥𝐵𝑦 ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))
3		dfima2 5971	. 2 ⊢ (𝐵 “ 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑥𝐵𝑦}
4	2, 3	elab2g 3611	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   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:  elima  5974  fvelima  6835  fvelimad  6836  opelco3  33749  fvelima2  42806  afvelima  44659