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Theorem fvelima2 41885
 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 5904 . . . 4 (𝐵 ∈ (𝐹𝐶) → (𝐵 ∈ (𝐹𝐶) ↔ ∃𝑥𝐶 𝑥𝐹𝐵))
21ibi 270 . . 3 (𝐵 ∈ (𝐹𝐶) → ∃𝑥𝐶 𝑥𝐹𝐵)
3 df-rex 3115 . . 3 (∃𝑥𝐶 𝑥𝐹𝐵 ↔ ∃𝑥(𝑥𝐶𝑥𝐹𝐵))
42, 3sylib 221 . 2 (𝐵 ∈ (𝐹𝐶) → ∃𝑥(𝑥𝐶𝑥𝐹𝐵))
5 fnbr 6434 . . . . . . . . 9 ((𝐹 Fn 𝐴𝑥𝐹𝐵) → 𝑥𝐴)
65adantrl 715 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥𝐴)
7 simprl 770 . . . . . . . 8 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥𝐶)
86, 7elind 4124 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → 𝑥 ∈ (𝐴𝐶))
9 fnfun 6427 . . . . . . . . 9 (𝐹 Fn 𝐴 → Fun 𝐹)
10 funbrfv 6695 . . . . . . . . . 10 (Fun 𝐹 → (𝑥𝐹𝐵 → (𝐹𝑥) = 𝐵))
1110imp 410 . . . . . . . . 9 ((Fun 𝐹𝑥𝐹𝐵) → (𝐹𝑥) = 𝐵)
129, 11sylan 583 . . . . . . . 8 ((𝐹 Fn 𝐴𝑥𝐹𝐵) → (𝐹𝑥) = 𝐵)
1312adantrl 715 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → (𝐹𝑥) = 𝐵)
148, 13jca 515 . . . . . 6 ((𝐹 Fn 𝐴 ∧ (𝑥𝐶𝑥𝐹𝐵)) → (𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
1514ex 416 . . . . 5 (𝐹 Fn 𝐴 → ((𝑥𝐶𝑥𝐹𝐵) → (𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵)))
1615eximdv 1918 . . . 4 (𝐹 Fn 𝐴 → (∃𝑥(𝑥𝐶𝑥𝐹𝐵) → ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵)))
1716imp 410 . . 3 ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥𝐶𝑥𝐹𝐵)) → ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
18 df-rex 3115 . . 3 (∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵 ↔ ∃𝑥(𝑥 ∈ (𝐴𝐶) ∧ (𝐹𝑥) = 𝐵))
1917, 18sylibr 237 . 2 ((𝐹 Fn 𝐴 ∧ ∃𝑥(𝑥𝐶𝑥𝐹𝐵)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
204, 19sylan2 595 1 ((𝐹 Fn 𝐴𝐵 ∈ (𝐹𝐶)) → ∃𝑥 ∈ (𝐴𝐶)(𝐹𝑥) = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112  ∃wrex 3110   ∩ cin 3883   class class class wbr 5033   “ cima 5526  Fun wfun 6322   Fn wfn 6323  ‘cfv 6328 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336 This theorem is referenced by:  limsupresxr  42395  liminfresxr  42396  liminfvalxr  42412
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