Theorem fvelima 5640

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 5032	. . . 4 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝐴 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴))
2	1	ibi 176	. . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴)
3		funbrfv 5627	. . . 4 ⊢ (Fun 𝐹 → (𝑥𝐹𝐴 → (𝐹‘𝑥) = 𝐴))
4	3	reximdv 2608	. . 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 1373   ∈ wcel 2177  ∃wrex 2486   class class class wbr 4048   “ cima 4683  Fun wfun 5271  ‘cfv 5277

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-pow 4223  ax-pr 4258

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3001  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-br 4049  df-opab 4111  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fv 5285

This theorem is referenced by:  ssimaex  5650  ctssdccl  7225  suplocexprlemmu  7844  suplocexprlemloc  7847  ennnfonelemex  12835