Theorem elimag 5013

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 4037	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝐴))
2	1	rexbidv 2498	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑥𝐵𝑦 ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))
3		dfima2 5011	. 2 ⊢ (𝐵 “ 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑥𝐵𝑦}
4	2, 3	elab2g 2911	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∃wrex 2476   class class class wbr 4033   “ cima 4666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676

This theorem is referenced by:  elima  5014  elrelimasn  5035  fvelima  5612  ecexr  6597