Theorem elimag 6064

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 5111	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝐴))
2	1	rexbidv 3188	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑥𝐵𝑦 ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))
3		dfima2 6062	. 2 ⊢ (𝐵 “ 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑥𝐵𝑦}
4	2, 3	elab2g 3637	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   class class class wbr 5107   “ cima 5662

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672

This theorem is used by:  elima  6065  fvelima2  6934  fvelima  6947  fvelimad  6949  opelco3  36362  afvelima  48063  inisegn0a  49772