Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvelima2 Structured version   Visualization version   GIF version

Theorem fvelima2 39789
 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 id 22 . . . 4 (𝐵 ∈ (𝐹𝐶) → 𝐵 ∈ (𝐹𝐶))
2 elimag 5505 . . . 4 (𝐵 ∈ (𝐹𝐶) → (𝐵 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 𝑥𝐹𝐵))
31, 2mpbid 222 . . 3 (𝐵 ∈ (𝐹𝐶) → ∃𝑥𝐶 𝑥𝐹𝐵)
4 df-rex 2947 . . 3 (∃𝑥𝐶 𝑥𝐹𝐵 ↔ ∃𝑥(𝑥𝐶𝑥𝐹𝐵))
53, 4sylib 208 . 2 (𝐵 ∈ (𝐹𝐶) → ∃𝑥(𝑥𝐶𝑥𝐹𝐵))
6 fnbr 6031 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐹𝐵) → 𝑥𝐴)
76adantrl 752 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥𝐴)
8 simprl 809 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥𝐶)
97, 8elind 3831 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥 ∈ (𝐴𝐶))
10 fnfun 6026 . . . . . . . . 9 (𝐹 Fn 𝐴 → Fun 𝐹)
11 funbrfv 6272 . . . . . . . . . 10 (Fun 𝐹 → (𝑥𝐹𝐵 → (𝐹𝑥) = 𝐵))
1211imp 444 . . . . . . . . 9 ((Fun 𝐹𝑥𝐹𝐵) → (𝐹𝑥) = 𝐵)
1310, 12sylan 487 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝐵) → (𝐹𝑥) = 𝐵)
1413adantrl 752 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → (𝐹𝑥) = 𝐵)
159, 14jca 553 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → (𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
1615ex 449 . . . . 5 (𝐹 Fn 𝐴 → ((𝑥𝐶𝑥𝐹𝐵) → (𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵)))
1716eximdv 1886 . . . 4 (𝐹 Fn 𝐴 → (∃𝑥(𝑥𝐶𝑥𝐹𝐵) → ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵)))
1817imp 444 . . 3 ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥𝐶𝑥𝐹𝐵)) → ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
19 df-rex 2947 . . 3 (∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
2018, 19sylibr 224 . 2 ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥𝐶𝑥𝐹𝐵)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
215, 20sylan2 490 1 ((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃wrex 2942   ∩ cin 3606   class class class wbr 4685   “ cima 5146  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934 This theorem is referenced by:  limsupresxr  40316  liminfresxr  40317  liminfvalxr  40333
 Copyright terms: Public domain W3C validator