Theorem fvelima 5583

Description: Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
fvelima	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fvelima

Step	Hyp	Ref	Expression
1		elimag 4989	. . . 4 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝐴 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴))
2	1	ibi 176	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴)
3		funbrfv 5570	. . . 4 ⊢ (Fun 𝐹 → (𝑥𝐹𝐴 → (𝐹‘𝑥) = 𝐴))
4	3	reximdv 2591	. . 3 ⊢ (Fun 𝐹 → (∃𝑥 ∈ 𝐵 𝑥𝐹𝐴 → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴))
5	2, 4	syl5 32	. 2 ⊢ (Fun 𝐹 → (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴))
6	5	imp 124	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2160  ∃wrex 2469   class class class wbr 4018   “ cima 4644  Fun wfun 5225  ‘cfv 5231

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fv 5239

This theorem is referenced by:  ssimaex  5593  ctssdccl  7128  suplocexprlemmu  7735  suplocexprlemloc  7738  ennnfonelemex  12433