Theorem imaexg 5138

Description: The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)

Assertion

Ref	Expression
imaexg	⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)

Proof of Theorem imaexg

Step	Hyp	Ref	Expression
1		imassrn 5135	. 2 ⊢ (𝐴 “ 𝐵) ⊆ ran 𝐴
2		rnexg 5045	. 2 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V)
3		ssexg 4270	. 2 ⊢ (((𝐴 “ 𝐵) ⊆ ran 𝐴 ∧ ran 𝐴 ∈ V) → (𝐴 “ 𝐵) ∈ V)
4	1, 2, 3	sylancr 418	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  ran crn 4773   “ cima 4775

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-xp 4778  df-cnv 4780  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785

This theorem is referenced by:  imaex  5139  ecexg  6805  fopwdom  7130  isinfinf  7195  ballotfilemrval  13244  isunitd  14396