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Theorem fvelima2 6961
Description: Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Assertion
Ref Expression
fvelima2 ((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹

Proof of Theorem fvelima2
StepHypRef Expression
1 elimag 6082 . . . 4 (𝐵 ∈ (𝐹𝐶) → (𝐵 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 𝑥𝐹𝐵))
21ibi 267 . . 3 (𝐵 ∈ (𝐹𝐶) → ∃𝑥𝐶 𝑥𝐹𝐵)
3 df-rex 3071 . . 3 (∃𝑥𝐶 𝑥𝐹𝐵 ↔ ∃𝑥(𝑥𝐶𝑥𝐹𝐵))
42, 3sylib 218 . 2 (𝐵 ∈ (𝐹𝐶) → ∃𝑥(𝑥𝐶𝑥𝐹𝐵))
5 fnbr 6676 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐹𝐵) → 𝑥𝐴)
65adantrl 716 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥𝐴)
7 simprl 771 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥𝐶)
86, 7elind 4200 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥 ∈ (𝐴𝐶))
9 fnfun 6668 . . . . . . . . 9 (𝐹 Fn 𝐴 → Fun 𝐹)
10 funbrfv 6957 . . . . . . . . . 10 (Fun 𝐹 → (𝑥𝐹𝐵 → (𝐹𝑥) = 𝐵))
1110imp 406 . . . . . . . . 9 ((Fun 𝐹𝑥𝐹𝐵) → (𝐹𝑥) = 𝐵)
129, 11sylan 580 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝐵) → (𝐹𝑥) = 𝐵)
1312adantrl 716 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → (𝐹𝑥) = 𝐵)
148, 13jca 511 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → (𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
1514ex 412 . . . . 5 (𝐹 Fn 𝐴 → ((𝑥𝐶𝑥𝐹𝐵) → (𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵)))
1615eximdv 1917 . . . 4 (𝐹 Fn 𝐴 → (∃𝑥(𝑥𝐶𝑥𝐹𝐵) → ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵)))
1716imp 406 . . 3 ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥𝐶𝑥𝐹𝐵)) → ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
18 df-rex 3071 . . 3 (∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
1917, 18sylibr 234 . 2 ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥𝐶𝑥𝐹𝐵)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
204, 19sylan2 593 1 ((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wrex 3070  cin 3950   class class class wbr 5143  cima 5688  Fun wfun 6555   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  exsslsb  33647  limsupresxr  45781  liminfresxr  45782  liminfvalxr  45798
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