Theorem elimag 5048

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 4066	. . 3 ⊢ (𝑦 = 𝐴 → (𝑥𝐵𝑦 ↔ 𝑥𝐵𝐴))
2	1	rexbidv 2511	. 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 ∈ 𝐶 𝑥𝐵𝑦 ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))
3		dfima2 5046	. 2 ⊢ (𝐵 “ 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐶 𝑥𝐵𝑦}
4	2, 3	elab2g 2930	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐶 𝑥𝐵𝐴))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1375   ∈ wcel 2180  ∃wrex 2489   class class class wbr 4062   “ cima 4699

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-xp 4702  df-cnv 4704  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709

This theorem is referenced by:  elima  5049  elrelimasn  5070  fvelima  5658  ecexr  6655