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Theorem fvelima2 43495
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 6018 . . . 4 (𝐵 ∈ (𝐹𝐶) → (𝐵 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 𝑥𝐹𝐵))
21ibi 267 . . 3 (𝐵 ∈ (𝐹𝐶) → ∃𝑥𝐶 𝑥𝐹𝐵)
3 df-rex 3075 . . 3 (∃𝑥𝐶 𝑥𝐹𝐵 ↔ ∃𝑥(𝑥𝐶𝑥𝐹𝐵))
42, 3sylib 217 . 2 (𝐵 ∈ (𝐹𝐶) → ∃𝑥(𝑥𝐶𝑥𝐹𝐵))
5 fnbr 6611 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐹𝐵) → 𝑥𝐴)
65adantrl 715 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥𝐴)
7 simprl 770 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥𝐶)
86, 7elind 4155 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥 ∈ (𝐴𝐶))
9 fnfun 6603 . . . . . . . . 9 (𝐹 Fn 𝐴 → Fun 𝐹)
10 funbrfv 6894 . . . . . . . . . 10 (Fun 𝐹 → (𝑥𝐹𝐵 → (𝐹𝑥) = 𝐵))
1110imp 408 . . . . . . . . 9 ((Fun 𝐹𝑥𝐹𝐵) → (𝐹𝑥) = 𝐵)
129, 11sylan 581 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝐵) → (𝐹𝑥) = 𝐵)
1312adantrl 715 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → (𝐹𝑥) = 𝐵)
148, 13jca 513 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → (𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
1514ex 414 . . . . 5 (𝐹 Fn 𝐴 → ((𝑥𝐶𝑥𝐹𝐵) → (𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵)))
1615eximdv 1921 . . . 4 (𝐹 Fn 𝐴 → (∃𝑥(𝑥𝐶𝑥𝐹𝐵) → ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵)))
1716imp 408 . . 3 ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥𝐶𝑥𝐹𝐵)) → ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
18 df-rex 3075 . . 3 (∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
1917, 18sylibr 233 . 2 ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥𝐶𝑥𝐹𝐵)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
204, 19sylan2 594 1 ((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wrex 3074  cin 3910   class class class wbr 5106  cima 5637  Fun wfun 6491   Fn wfn 6492  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  limsupresxr  44014  liminfresxr  44015  liminfvalxr  44031
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