Theorem fvelima 5608

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 5009	. . . 4 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝐴 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴))
2	1	ibi 176	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴)
3		funbrfv 5595	. . . 4 ⊢ (Fun 𝐹 → (𝑥𝐹𝐴 → (𝐹‘𝑥) = 𝐴))
4	3	reximdv 2595	. . 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 2164  ∃wrex 2473   class class class wbr 4029   “ cima 4662  Fun wfun 5248  ‘cfv 5254

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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fv 5262

This theorem is referenced by:  ssimaex  5618  ctssdccl  7170  suplocexprlemmu  7778  suplocexprlemloc  7781  ennnfonelemex  12571